Finite Atomized Semilattices
aa r X i v : . [ m a t h . R A ] F e b Finite Atomized Semilattices
Fernando Martin-Maroto and Gonzalo G. de Polavieja
Champalimaud Research, Champalimaud Centre for the Unknown, Lisbon, Portugal Algebraic AI, Madrid, Spain
February 17, 2021
Abstract
We show that every finite semilattice can be represented as an atomized semilattice, analgebraic structure with additional elements (atoms) that extend the semilattice’s partialorder. Each atom maps to one subdirectly irreducible component, and the set of atomsforms a hypergraph that fully defines the semilattice. An atomization always exists andis unique up to “redundant atoms”. Atomized semilattices are representations that canbe used as computational tools for building semilattice models from sentences, as well asbuilding its subalgebras and products. Atomized semilattices can be applied to machinelearning and to the study of semantic embeddings into algebras with idempotent operators. ontents Introduction
Elements in a semilattices, just like in Boolean algebras, can be represented as sets [1]. Weleverage this fact to extend the semilattice to an algebra of two sorts, the atomized semilattice,which is well suited as a computational tool to introduce additional properties in models aswell as to carry out selected properties from model to model.Our main results are the concept of atomization, redundancy and the full crossing con-struction which allows building semilattices from a set of positive atomic sentences. Specialattention is paid to relative freedom between models as it may play a role in the field of machinelearning. We provide a few examples to illustrate the power of the atomized semilattice andexplore its most basic properties.Although this manuscript does not explicitly discuss machine learning, it is meant as areference for researchers interested in the mathematics of algebraic machine learning [2]. Wedeal here with the purely algebraic aspects of the atomised semilattice construction, which mayalso be of interest to algebraists.In section 2 we give an axiomatization for atomized semilattices, introduce the conceptsof redundant atom and discriminant atom and also extend the semilattice homomorphism toatomized semilattices. We discuss some representational choices that endow the elements ofan atomized semilattice, including atoms, with the ”universal” property that allows to identifythem across every model generated by the same constants.In section 3 we start by introducing the notion of relative algebraic freedom (definition3.1). We prove that redundant atoms can be discarded (theorem 9) and that finite semilatticescan be represented with a unique atomization without redundant atoms (theorem 13). Wecontinue with the central concept of full crossing that permits building the freest model of aset of atomic sentences (theorem 16).Section 4 is a selection of results. We show that some subalgebras can be constructeddirectly from the restriction of the atoms (theorem 22), and later show how to calculate anysubalgebra in a simple way (theorem 33). In theorem 27 we extend the link (introduced intheorem 15) between freedom and redundancy. We explain how to build congruences using fullcrossing (theorem 29) and use the congruences to build join models, subalgebras and products.We provide a method to calculate join models, which are the result of gluing together varioussemilattice models (see theorem 31), and discuss when it is possible to find embeddings fromtwo semilattices into their join model (theorem 32). Products of models are considered intheorem 36. Finally, the connection between the atoms and the irreducible components of asubdirect product (theorem 37) is made explicit.3
Preliminaries
A semilattice [3] is an algebra with a single binary operation ⊙ that is commutative, associativeand idempotent. The idempotent operator induces a relationship of (partial) order. We say a ≤ b in a semilattice M if and only if M | = ( b = b ⊙ a ). The elements of a semilattice form apartially ordered set. We often refer to semilattices as semilattice models or, simply, models.A semilattice model “over a set of constants C ” is assumed to be generated by its constants,i.e. every element can be found as an idempotent summation of constants. We only deal herewith finite models so we assume C is a finite set. In order to get a model generated by itsconstants, if an element b of a semilattice model is not equal to any idempotent summation ofconstants then it is possible to make b a constant. Without loss of generality we assume in thistext that every finite semilattice is generated by its constants.Given a set C of constants, the free algebra [3] F C ( ∅ ) contains a different element for eachconstant in C and for each idempotent summation of constants in C , where two summationsare considered equal if and only if they can be proved equal using the commutative, associativeand idempotent properties of ⊙ . From these three properties follows easily that the cardinalityof the freest model (the free algebra) is 2 | C | − F C ( ∅ ) is the well-know term algebra. Theelements of F C ( ∅ ) are called “terms”.There is always a natural homomorphism [3] ν M from the term algebra F C ( ∅ ) onto anymodel M that maps terms to elements of M . The inverse image of the natural homomorphism ν M : F C ( ∅ ) → M partitions the terms over C into as many equivalence classes as elements arein M . We can see the terms in each equivalence class of F C ( ∅ ) as an alternative name for thesame element of M . Each semilattice model over C is a partition of the set F C ( ∅ ). Since theelements of a semilattice are equivalence classes defined in F C ( ∅ ), every semilattice over C havefewer than 2 | C | − F C ( ∅ ), names such as t or s for whichit may happen in some model M that M | = ( ν M ( t ) = ν M ( s )) or, simply, M | = ( t = s ) as wedo not explicitly write the homomorphism either. The reason for this unusual choice is thatby using terms we can refer to the “same element” in different models. Atomized semilatticeswere introduced in the context of machine learning, where we do not stay working with thesame model for long, instead, when new data challenges the model, the model gets replaced byanother one. Using names for elements of particular models would force us to build explicitmappings when models are replaced and that turns out to be an unnecessary complication.A “duple” r = ( r L , r R ) is defined in this text as an ordered pair of elements of F C ( ∅ ) (terms).Positive and negative duples, written r + and r − , are used as follows: we say M | = r + if and4nly if M | = ( ν M ( r L ) ≤ ν M ( r R )) and we say M | = r − if and only if M = ( ν M ( r L ) ≤ ν M ( r R ))or, without writing natural map symbol M | = ( r L ≤ r R ).A semilattice model M is atomized when it is extended with a set of additional elementswe call “atoms”. Each semilattice is a partial order and an atomization of the semilattice is anextension of this partial order. However, an atomized semilattice is not a semilattice extension;in atomized semilattices the idempotent operator is defined exclusively for regular elementswhile the order relation is defined for all, atoms and regular elements. If a and b are regularelements and φ is an atom, we can say a ⊙ b and φ < a but not a ⊙ φ . The regular elements ofan atomized semilattice form a semilattice. All the elements of an atomized semilattice form apartially ordered set. Definition 2.1.
An atomized semilattice over a set of constants C is a structure M withelements of two sorts, the regular elements { a, b, c, ... } and the atoms { φ, ψ, ... } , with an idem-potent, commutative and associative binary operator ⊙ defined for regular elements and apartial order relation < (i.e. a binary, reflexive, antisymmetric and transitive relation) that isdefined for all the elements, such that the regular elements are either constants or idempotentsummations of constants, and M satisfies the axioms of the operations and the additional: ∀ φ ∃ c : ( c ∈ C ) ∧ ( φ < c ) , (AS1) ∀ φ ∀ a ( a φ ) , (AS2) ∀ a ∀ b ( a ≤ b ⇔ ¬∃ φ : (( φ < a ) ∧ ( φ < b ))) , (AS3) ∀ φ ∀ a ∀ b ( φ < a ⊙ b ⇔ ( φ < a ) ∨ ( φ < b )) , (AS4) ∀ c ∈ C (( φ < c ) ⇔ ( ψ < c )) ⇒ ( φ = ψ ) , (AS5) ∀ a ∃ φ : ( φ < a ) . (AS6)We use Greek letters to represent atoms and Latin letters to represent regular elements.Constants in C are considered regular elements. We use C ( t ) for the component constants ofa term t (the constants mentioned in the term), and L aM ( b ) for the “lower atomic segment” ofan element b in the model M which is the set of the atoms φ that satisfy M | = ( φ < b ) (seesection 6 for notation and definitions). The “upper constant segment” U c ( φ ) of an atom φ isthe set of constants c ∈ C that satisfy φ < c in the model M . We drop the subindex M andwrite U c ( φ ) instead of U cM ( φ ) for reasons that will become apparent soon.We are going to show first that the partial order of the atomized semilattice coincides withthat of the semilattice spawned by its regular elements: ∀ a ∀ b ( a ≤ b ⇔ a ⊙ b = b ) . (AS3b)5 heorem 1. Assume AS4 and the antisymmetry of the order relation.i) AS3 implies AS3b.ii) Assume ∀ a ∀ b ( ∀ φ (( φ < a ) ⇔ ( φ < b )) ⇒ ( a = b )) . Then AS b ⇒ AS . iii) AS3 implies ∀ a ∀ b ( ∀ φ (( φ < a ) ⇔ ( φ < b )) ⇒ ( a = b )) .Proof. (i) AS ⇒ AS b : Assume a ≤ b . AS3 implies ¬∃ φ : (( φ < a ) ∧ ( φ < b )) and then( φ < a ) ∨ ( φ < b ) ⇔ ( φ < b ). From here, using AS4, follows ( φ < a ⊙ b ) ⇔ ( φ < b ), and we canuse AS3 (right to left this time) to get ( b ≤ a ⊙ b ) ∧ ( a ⊙ b ≤ b ), and from the antisymmetry ofthe order relation, i.e. ( x = y ) ⇔ ( x ≤ y ) ∧ ( y ≤ x ), we obtain a = a ⊙ b .Assume a ⊙ b = b . Then ( φ < a ⊙ b ) ⇔ ( φ < b ), and using AS4: ( φ < a ) ∨ ( φ < b ) ⇔ ( φ < b )which implies ¬∃ φ : (( φ < a ) ∧ ( φ < b )) from which, using AS3, we get a ≤ b .(ii) AS b ⇒ AS
3: Assume a ≤ b . AS3b implies a ⊙ b = b . From AS4 we get that ( φ < a ) ∨ ( φ
Let t, s ∈ F C ( ∅ ) be two terms that represent two regular elements ν M ( t ) and ν M ( s ) of an atomized model M over a finite set of constants C . Let φ be an atom, c a constantin C and let a be a regular element of M :i) ∀ t ∀ c ( c ∈ C ( t ) ⇒ ν M ( c ) ≤ ν M ( t )) ,ii) φ < ν M ( t ) ⇔ ∃ c : (( c ∈ C ( t )) ∧ ( φ < ν M ( c ))) ,iii) ( φ < a ) ⇔ ∃ c : (( c ∈ C ) ∧ ( φ < ν M ( c ) ≤ a )) ,iv) L aM ( ν M ( t )) = { φ ∈ M : C ( t ) ∩ U c ( φ ) = ∅} , v) L aM ( ν M ( s ) ⊙ ν M ( t )) = L aM ( ν M ( t )) ∪ L aM ( ν M ( s )) ,vi) ν M ( t ) ≤ ν M ( s ) ⇔ L aM ( ν M ( t )) ⊆ L aM ( ν M ( s )) . roof. (i) From t = t ⊙ c and the natural homomorphism ν M ( t ) = ν M ( t ⊙ c ) = ν M ( t ) ⊙ ν M ( c )we get ν M ( c ) ≤ ν M ( t ).(ii) Right to left, φ < ν M ( c ) ≤ ν M ( t ) follows from i and, from and the transitivity of the orderrelation, φ < ν M ( t ). Left to right can be proven from the fourth axiom of atomized models φ < a ⊙ b ⇒ ( φ < a ) ∨ ( φ < b ) applied to the component constants C ( t ) of t . The number ofcomponent constants of t is at least 1 and at most | C | so it is a finite number and we need toapply this axiom a finite number of times to get φ < ν M ( c ) for some component constant of t .This proves (ii), and (iii) is a consequence of (ii) that follows with just choosing any term t of F C ( ∅ ) to represent element a = ν M ( t ).(iv) Consider an atom φ ∈ L aM ( ν M ( t )) then φ < ν M ( t ) and from proposition (ii) there isa component constant c ∈ C ( t ) such that φ < ν M ( c ) which means that c ∈ C ( t ) ∩ U c ( φ ).Conversely, if c ∈ C ( t ) ∩ U c ( φ ) then φ < ν M ( c ) ≤ ν M ( t ) and φ ∈ L aM ( ν M ( t )).(v) Since ν M is a homomorphism ν M ( s ) ⊙ ν M ( t ) = ν M ( s ⊙ t ) and, using proposition (iv) L aM ( ν M ( s ⊙ t )) = { φ ∈ M : C ( s ⊙ t ) ∩ U c ( φ ) = ∅} = { φ ∈ M : ( C ( s ) ∪ C ( t )) ∩ U c ( φ ) = ∅} = L aM ( ν M ( t )) ∪ L aM ( ν M ( s )). Note that this proposition is an alternative way to write axiom AS4.(vi) It is straightforward from proposition v and AS3b or AS3.The first axiom of atomized semilattices says that the upper constant segment of an atomis never empty while the fifth axiom says that two distinct atoms cannot have the same upperconstant segment.Axiom AS3 implies that the order relation for regular elements is encoded and fully de-termined by the atoms of the model. Theorem 2 propositions (iv) to (vi) show that it sufficeswith knowing the constants in the upper segments of each atom in the atomized model to knowthe entire model. Furthermore, it is enough with knowing the atoms of a model M and thecomponent constants C ( t ) of a term to know the atoms in the lower segment of t in M .Theorem 2 proposition (v) proves the linearity property: L aM ( s ⊙ t ) = L aM ( t ) ∪ L aM ( s ) , where we have omitted the natural homomorphism.Theorem 2 proposition (vi) and AS3b imply that M | = ( t = s ) ⇔ L aM ( t ) = L aM ( s ) . Therefore, it does not matter which term we use to represent a regular element we get the sameatoms using: L aM ( t ) = { φ ∈ M : C ( t ) ∩ U c ( φ ) = ∅} .
7n order to use U c ( φ ) instead of U cM ( φ ) we just need to assume, without loss of generality,that atoms of different atomized semilattice models that have the same upper constant segmentsalso have the same name. With this choice an atom can be defined by a set U c ( φ ), independentlyof other atoms and independently of the model it belongs to. We can define the atoms by theirupper constant segments U c ( φ ) as well as the models by the atoms they have, as follows: Definition 2.2.
Consider a non-empty subset U c ( φ ) of a set of constants C . For any atomizedsemilattice M over C , any set A of atoms that atomizes M and any constant c in C , the atom φ defined by U c ( φ ) is an atom that satisfies M | = ( φ < c ) if and only if φ ∈ A and c ∈ U c ( φ ).Since atoms are non-empty sets of constants any atomization of a semilattice model is ahypergraph with the atoms as hyperedges and the constants as vertexes.Axiom AS3 can be rewritten to show the connection between universally defined atomsand models as: ( t ≤ M s ) ⇔ ¬∃ φ : (( φ ∈ M ) ∧ ( φ < t ) ∧ ( φ < s )) . where t and s are terms and t ≤ M s means M | = ( t ≤ s ) and the natural homomorphism hasbeen omitted. We add the subscript in ≤ M to highlight the difference with < that comparesatoms and terms independently of the model. We do not use subscripts and simply write ≤ for ≤ M when there is no ambiguity.The next theorem permits to identify U c ( φ ) with the upper constant segment of φ in thefreest model F C ( ∅ ), as well as ≤ with ≤ F C ( ∅ ) .We sometimes use [ A ] to refer to the model atomized by a set of atoms A , and sometimesuse the same letter M to refer to both, the model and the set of atoms. Theorem 3. If A is a set of atoms that atomizes F C ( ∅ ) and ψ is any atom, the set A ∪ { ψ } isalso an atomization for F C ( ∅ ) .Proof. Let t , s be two terms. In any semilattice model, C ( t ) ⊆ C ( s ) implies t ≤ s . The freestsemilattice is characterized by the double implication: t ≤ s if and only if C ( t ) ⊆ C ( s ).An atom defined by its non-empty set U c ( φ ) according to definition 2.2 necessarily satisfiesthe axioms AS1 and is consistent with axioms AS2, AS4 and AS5. Since [ A ] satisfies axiomAS6 then [ A ∪ { ψ } ] also satisfies axiom AS6. Consider the axiom AS3: ∀ a ∀ b ( a ≤ b ⇔ ¬∃ φ :(( φ < a ) ∧ ( φ < b ))). If C ( t ) ⊆ C ( s ), from theorem 2(iv) follows that ( ψ < t ) ⇒ ( ψ < s ). If C ( t ) C ( s ) then F C ( ∅ ) = [ A ] | = ( t < s ) and the clause ¬∃ φ : (( φ < t ) ∧ ( φ < s ))) does not hold.In both cases, C ( t ) ⊆ C ( s ) and C ( t ) C ( s ), if [ A ] satisfies the clause ¬∃ φ : (( φ < t ) ∧ ( φ < s )))8hen [ A ∪ { ψ } ] also does, so t ≤ s holds in [ A ∪ { ψ } ] if and only if it holds in [ A ] and it followsthat the semilattice formed by the regular elements of [ A ∪ { ψ } ] is equal to F C ( ∅ ).A corollary of theorem 3 is that the set of all possible atoms over C is an atomization of F C ( ∅ ). We can say that every atom “is in” the freest semilattice F C ( ∅ ) and write φ ∈ F C ( ∅ )for any atom φ defined by a proper subset of C . We will see later that there are atomizationsfor the freest semilattice with fewer atoms. Definition 2.3.
Let M = [ A ] be the semilattice spawned by the atoms A . We say an atom φ is in M , written φ ∈ M , if the semilattice formed by the regular elements of [ A ∪ { φ } ] is equalto M . Definition 2.4.
Let M be a semilattice. We say an atom φ is in M , written φ ∈ M , if there isan atomized semilattice spawned by some set of atoms that contains φ with regular elementsforming a semilattice equal to M .Since an atom is universally defined by a non-empty subset of C , by using definition 2.2 itis possible to identify the same atom in different models and to compare the same atom withelements of different models provided that the models are generated by the same constants. Inthe next two theorems we compare the same atoms with different models: F C ( ∅ ), M and N . Theorem 4.
Let t, s ∈ F C ( ∅ ) be two terms and M an atomized semilattice model. Let φ be anatom and ν M : F C ( ∅ ) → M the natural homomorphism. i ) ( F C ( ∅ ) | = ( t ≤ s )) ⇒ M | = ( ν M ( t ) ≤ ν M ( s )) ii ) ( φ ∈ M ) ∧ ( F C ( ∅ ) | = ( φ < s )) ⇔ M | = ( φ < ν M ( s )) Proof. i) Follows immediately from the fact that ν M is a homomorphism and is provided herefor comparison with proposition (ii).ii) Note that we use here the same atom φ in the contexts of two atomized models, F C ( ∅ ) and M .Left to right: using theorem 2(iii), F C ( ∅ ) | = ( φ < s ) implies that there is some constant c suchthat F C ( ∅ ) | = ( φ < c ≤ s ) and then, from the natural homomorphism, M | = ( ν M ( c ) ≤ ν M ( s )).From definition 2.2, F C ( ∅ ) | = ( φ < c ) requires c ∈ U c ( φ ) and then, because we assume φ ∈ M ,the same definition implies M | = ( φ < ν M ( c )). Using the transitive property of the orderrelation M | = ( φ < ν M ( s )).Right to left can be proven with the same argument than left to right interchanging the rolesof M and F C ( ∅ ) except for the fact that we do not need to require φ ∈ F C ( ∅ ) as this is alwaysthe case for any atom, as proven in theorem 3.9ote that the implication arrow in proposition (i) of the last theorem goes only left toright while in proposition (ii) goes in both directions.We say that an atom φ “discriminates” a duple r = ( r L , r R ) if ( φ < r L ) ∧ ( φ < r R ). If φ ∈ M then axiom AS3 implies that M | = r − . Theorem 5.
Let t, s ∈ F C ( ∅ ) be two terms and M and N two models. If an atom φ discrimi-nates a duple ( t, s ) in M then, either φ N or φ also discriminates ( t, s ) in N .Proof. If φ discriminates ( t, s ) in M , using part (ii) of theorem 4 right-to-left for both terms,follows that φ discriminates ( t, s ) in F C ( ∅ ) and then, using again theorem 4(ii) left-to-right, weget that either φ N or φ also discriminates ( t, s ) in N . Definition 2.5.
We say an atom φ is wider than atom η if it is different than η and for everyconstant c , ( η < c ) ⇒ ( φ < c ).Equivalently, an atom φ is wider than atom η when U c ( η ) ( U c ( φ ). It is possible tofurther extend the partial order by taking φ < η if and only if φ is wider than η . In this textatoms are only compared with regular elements and such extension is not needed. Definition 2.6.
We say an atom φ is redundant with M when for each constant c such that φ < c there is at least one atom η < c in M such that φ is wider than η .With respect to a model M we say that an atom φ is “external to M ” if φ is not in M (asdefined in 2.3) and we write φ M . If an atom is in M it may be either redundant with M ornot. If the atom is in M and is not redundant with M then we call it a “non-redundant atomin M ”. Therefore, with respect to a model M an atom can be either a “non-redundant atomin M ” or “redundant with M ” or “external to M ”.We will see in theorem 9 that redundant atoms can be discarded without altering thesemilattice model. In addition, in theorem 13 we prove that the set of atoms that are non-redundant with M is unique, and there is only one atomization of M with every atom non-redundant in M . As a consequence of these theorems the non-redundant atoms in M are thosethat are on every possible atomization of M while the redundant atoms with M are thosethat are only in some atomizations of M . When we work with an specific set of atoms A , aredundant atom with M may or may not be in set A .10tomized semilattice models often have many atoms. We are interested in the subset ofnon-redundant atoms that suffices to atomize the model. For example, in theorem 18 we willshow that F C ( ∅ ) can be atomized with as few as | C | non-redundant atoms, however, every oneof the 2 | C | − C is an atom in F C ( ∅ ) as proven in theorem 3. The numberof atoms in a model M = [ A ] atomized by a set A of atoms is usually much larger than | A | asmost atoms might be redundant.The zero atom, ⊖ c , is defined by U c ( ⊖ c ) = C and satisfies ⊖ c < d for each constant d ∈ C .We can always add to any atomized model the atom ⊖ c without changing the semilattice formedby its regular elements. If a set of atoms A spawns a model [ A ] then the semilattice spawnedby [ A ∪ {⊖ c } ] is the same semilattice than [ A ].If we use definition 2.2 a set A of atoms always forms an atomized semilattice if it satisfiesthe sixth axiom. If A does not form and atomized semilattice then theorem 7 tells us that A ∪ {⊖ c } does.Atom ⊖ c is redundant in most models. It is redundant unless there is at least one constant q such that M | = ∀ x ( q ≤ x ), a constant that acts as a neutral element. If there are two suchconstants, q and q , then M | = ( q = q ). Theorem 6.
Let M be a semilattice over C .i) ⊖ c is in M ,ii) ⊖ c is a non-redundant atom in M if and only if there is at least one constant q ∈ C suchthat M | = ∀ x ( q ≤ x ) .Proof. (i) Suppose M is atomized by a set of atoms A . Since F C ( ∅ ) | = ∀ t ( ⊖ c < t ) then, fromtheorem 4(ii), the model spawned by A ∪ {⊖ c } satisfies ∀ t ( ⊖ c < t ). Since an atom in the lowersegment of every regular element cannot discriminate any duple, then [ A ∪ {⊖ c } ] = M and itfollows that ⊖ c ∈ M .(ii) By definition, ⊖ c is redundant with M if for each constant c ∈ C there is at least one atom η ∈ M different than ⊖ c such that η < c . It follows that ⊖ c is not redundant with M if andonly if there is some constant q ∈ C for which there is no other atom η ∈ M such that η < q .We have shown in proposition (i) that ⊖ c ∈ M so if ⊖ c is not redundant with M then ⊖ c isa non-redundant atom in M . Therefore, if ⊖ c is a non-redundant atom in M there is at leastone constant q that only has ⊖ c in its lower atomic segment. Since ⊖ c is in the lower segmentof every element of M , theorem 2(iii) implies that M | = ∀ x ( q ≤ x ).On the other direction, if there is a constant q such that M | = ∀ x ( q ≤ x ) then q ≤ c for eachconstant c . Every atom φ < q satisfies φ < c for every constant of C , therefore, U c ( φ ) = C andaxiom AS5 implies φ = ⊖ c . In addition, ⊖ c cannot be redundant with M because if it were,11here would be an atom η < q with ⊖ c wider than η contradicting that q only has ⊖ c in itslower segment. Theorem 7.
Let A be a set of atoms over the constants C with each atom defined by its upperconstant segment according to 2.2. Either A or A ∪ {⊖ c } spawns an atomized semilattice model.Proof. To construct the atomized semilattice model M = [ A ] spawned by a set of atoms A , we start with the relation ≤ M over F C ( ∅ ) defined, for any two terms t, s ∈ F C ( ∅ ), by( t ≤ M s ) ≡ ¬∃ φ : ( φ ∈ A )(( φ < t ) ∧ ( φ < s )), where ( φ < t ) ≡ ( C ( t ) ∩ U c ( φ ) = ∅ ). Theelements of M correspond to equivalence classes in F C ( ∅ ) with t and s in the same class if andonly if ( t ≤ M s ) ∧ ( s ≤ M t ). The natural homorphism ν M ( t ) maps each term to its equivalenceclass and ( ν M ( t ) = ν M ( s )) ≡ ( t ≤ M s ) ∧ ( s ≤ M t ). The idempotent summation a ⊙ b for twoelements a and b of M can be built by choosing any terms in the inverse images of the naturalhomorphism, say t a and t b with a = ν M ( t a ) and b = ν M ( t b ) and by calculating ν M ( t a ⊙ t b ).With the help of the idempotent operator ⊙ we can extend the relation ≤ M defined for F C ( ∅ )to a relation in M using the usual M | = ( a ≤ b ) iff M | = ( b = a ⊙ b ).To show that the idempotent summation of elements is consistently defined, suppose we chooseother terms s a , s b such that a = ν M ( s a ) and b = ν M ( s b ). Then ( ν M ( t a ) = a = ν M ( s a )) and( ν M ( t b ) = b = ν M ( s b )) and these sentences are equivalent to ( t a ≤ M s a ) ∧ ( s a ≤ M t a ) and( t b ≤ M s b ) ∧ ( s b ≤ M t b ) respectively. Using the definition of ≤ M in F C ( ∅ ) given above it isstraightforward to show that these two sentences are true if and only if { φ : ( φ ∈ A ) ∧ ( φ Definition 3.1. Let A and B be two models over the constants C . We say that model A is“freer or as free” than model B if for every duple r over C we have B | = r − ⇒ A | = r − Theorem 8. Let C be a set of constants, K ⊆ C and r a duple over K . Let M be an atomizedsemilattice model over C and A a set of atoms that atomizes M . Let B be a subset of A and N = [ B ∪ {⊖ K } ] an atomized semilattice model over K .i) If M | = r + then N | = r + .ii) M is as free or freer than N .Proof. Let r = ( r L , r R ), the duple over K . Axiom AS3 of atomized semilattices in N reads ν N ( r L ) ≤ ν N ( r R ) ⇔ ¬∃ φ : (( φ ∈ B ∪ {⊖ K } ]) ∧ ( φ < r L ) ∧ ( φ < r R )) while, for model M is ν M ( r L ) ≤ ν M ( r R ) ⇔ ¬∃ φ : (( φ ∈ A ) ∧ ( φ < r L ) ∧ ( φ < r R )). Since ⊖ K discriminates no dupleover K and B ⊆ A , it follows that ( ν M ( r L ) ≤ ν M ( r R )) ⇒ ( ν N ( r L ) ≤ ν N ( r R )). Therefore, amodel N spawned by any subset of atoms of M satisfies all the positive duples satisfied by M .By negating the sentence above we get ( ν N ( r L ) ν N ( r R )) ⇒ ( ν M ( r L ) ν M ( r R )) which meansthat M satisfies all the negative duples of N and, hence, M is as free or freer than N .We need a few definitions:The discriminant dis M ( a, b ) in an atomized model M is the set L aM ( a ) − L aM ( b ) of atomsof M . From theorem 2(iv) follows that M | = a ≤ b if and only if dis M ( a, b ) is empty.For each atom φ = ⊖ c , let the “pinning term” T φ be equal to the idempotent summationof all the constants in the set C − U c ( φ ). For each constant c ∈ U c ( φ ), i.e, for each constantsuch that ( φ < c ), define the “pinning duple” as ¬ ( c < T φ ). P R ( φ ) is the set of pinning duplesof an atom φ . Note that pinning terms and pinning duples of an atom are independent of themodel as they only depend upon C and the set U c ( φ ).Let T h ( M ) be the set of duples satisfied by (the regular elements of) the model M , andthe positive and negative theories of M , written T h +0 ( M ) and T h − ( M ) respectively, are thesets of positive duples satisfied by M and the set of negative duples satisfied by M .The next theorem shows that an atom that is redundant with a model M atomized by aset of atoms A can be added to A or taken out from A without changing the model.14 heorem 9. Let A be a set of atoms that atomizes a model M and φ an atom of A . The set A − { φ } also atomizes M if and only if φ is redundant with M .Proof. Let T h +0 ( M ) be the set of all positive duples satisfied by the regular elements of M .Theorem 8 tells us that positive duples do not become negative when atoms are taken out ofan atomization. Taking out an atom φ from a model M produces a model N | = T h +0 ( M ).Therefore, when removing an atom we only need to worry about negative duples that maybecome positive. To prove that a redundant atom can be eliminated, let a and b be a pair ofterms of F C ( ∅ ) and suppose M | = ( a b ), a negative duple satisfied by M and discriminatedby an atom φ < c ≤ a where c is some constant. If φ is redundant there is an atom η < c in M such that φ is wider than η . Suppose η < b . There is a constant e such that η < e ≤ b .Because φ is wider than η then φ < e ≤ b which contradicts the assumption that φ ∈ dis M ( a, b ).Therefore, N | = ( η < b ), and then η ∈ dis N ( a, b ) and N | = ( a b ). It follows that any negativeduple of M is also negative in N and, if N models the same positive and negative duples than M , then the semilattices M and N are equal.Conversely, assume atom φ can be eliminated without altering M . Also assume φ = ⊖ c . Thepinning term T φ is the idempotent summation of all the constants in the set C − U c ( φ ). Foreach constant c such that φ < c we have φ ∈ dis M ( c, T φ ). From F C ( ∅ ) | = ( φ < T φ ) and theorem5 follows that φ discriminates duple ( c, T φ ) in M . If φ can be eliminated without altering M ,for each constant c in U c ( φ ) there should be some other atom η c in M with η c < c and η c = φ discriminating the duple ( c, T φ ) which implies T φ < T η c or, equivalently, that φ is wider than η c . Hence, for each constant such φ < c there is an η c ∈ M such that φ is wider than η c whichproves φ is redundant.Finally, suppose that φ = ⊖ c is a not-redundant atom in M . Then theorem 6 tells us thatthere is a constant q ∈ C such that M | = ∀ x ( q ≤ x ), a constant q with a lower atomic segmentthat contains only a single atom: φ (see the proof of 6). If φ is taken out of the model therewould be a constant with no atoms in its lower segment, which contradicts the sixth axiom ofatomized semilattices. Hence, A − { φ } does not atomize M . Theorem 10. Let M be a model and φ = ⊖ c a non-redundant atom of M . There is at leastone pinning duple that is discriminated by φ and only by φ .Proof. As in the proof of theorem 9, for each constant c ∈ U c ( φ ) we have φ ∈ dis M ( c, T φ )where T φ is the pinning term of φ . This is a consequence of theorem 5 and F C ( ∅ ) | = ( φ < T φ ).If, for a pinning duple ( c, T φ ) there is an atom ϕ = φ of M that discriminates ( c, T φ ) then φ iswider than ϕ and, if the same is true for every pinning duple in P R ( φ ), then φ is redundantwith M , which is against our assumptions. Therefore, there should be at least one constant c such that ( c, T φ ) is discriminated only by φ . 15e introduce now the union of atoms, φ ▽ ψ , that is an an atom with an upper constantsegment U c ( φ ▽ ψ ) = U c ( φ ) ∪ U c ( ψ ). Theorem 11. Let x be a regular element of an atomized semilattice model M and let φ , ψ and η be atoms of M. The union of atoms has the properties: i ) φ ▽ φ = φ,ii ) ▽ is commutative and associative ,iii ) φ < x ⇒ ( φ ▽ ψ < x ) ,iv ) ( φ ▽ ψ < x ) ⇔ ( φ < x ) ∨ ( ψ < x ) ,v ) ( φ ▽ ψ < x ) ∧ ( φ < x ) ⇒ ( ψ < x ) .vi ) φ is wider or equal to η if and only if φ = φ ▽ η. Proof. According to definition 2.2 an atom is determined by the constants in its upper segment,therefore atom φ ▽ ψ is fully defined by U c ( φ ▽ ψ ) = U c ( φ ) ∪ U c ( ψ ). Propositions i and iifollow from the idempotence, commutativity and associativity of the union of sets. Let t be aterm of F C ( ∅ ) and let ν M be the natural homomorphism of F C ( ∅ ) onto M . Theorem 2 saysthat L aM ( ν M ( t )) = { φ ∈ M : C ( t ) ∩ U c ( φ ) = ∅} , which permits to calculate the lower atomicsegment of an element ν M ( t ), represented with any term t , by using the component constantsof the term. Let t be any term such that x = ν M ( t ):(iii) φ < x implies that exists c ∈ C ( t ) such that φ < c and, hence, c ∈ U c ( φ ) ⊆ U c ( φ ▽ ψ ) so φ ▽ ψ < c ≤ x .(iv) Right to left, is the same than proposition (iii). Left to right; the left hand side can bewritten as ∃ t ∃ c (( x = ν M ( t )) ∧ c ∈ C ( t ) ∧ ( φ ▽ ψ < c )), which implies c ∈ U c ( φ ) ∪ U c ( ψ ) andthen ( φ < c ) ∨ ( ψ < c ) that, together with c ≤ x , yields ( φ < x ) ∨ ( ψ < x ).(v) is a trivial (but useful) consequence of proposition (iv).(vi) φ is wider than, or equal to, atom η if and only if for every constant c , ( η < c ) ⇒ ( φ < c )or, in other words, U c ( η ) ⊆ U c ( φ ). It follows that U c ( φ ▽ η ) = U c ( φ ) ∪ U c ( η ) = U c ( φ ). Hence, φ is wider than or equal to atom η if and only if φ = φ ▽ η . Theorem 12. Let M be an atomized model over a finite set C of constants. Let φ be an atomthat may or may not be in the atomization of M .i) φ is redundant with M if and only if is a union φ = ▽ i η i of atoms of M such that ∀ i ( φ = η i ) .ii) φ is redundant with M if and only if is a union of two or more non-redundant atoms of M . roof. If φ is redundant for each constant such thatφ < c there is an atom η i of M such that φ is wider than η i and η i < c . Theorem 11(vi) states that if φ is wider than η i then φ = φ ▽ η i and, since for each constant c ∈ U c ( φ ) there is some η i such that η i < c then U c ( φ ) = ∪ i U c ( η i ).It follows φ = ▽ i η i . Conversely if φ is a union of atoms, φ = ▽ i η i , then for each constant such φ < c there is some atom η i that contains c in its upper constant segment with φ wider than η i , and then φ is redundant with M . This proves (i).Since C is finite then | U c ( η i ) | < | U c ( φ ) | . If any of the atoms η i is redundant with M , then η i can also be expressed as a union of atoms of M . We can continue replacing the redundant atomsfor others with ever smaller upper constant segments until reaching non-redundant atoms of M .Because ▽ is associative there is at least one decomposition of φ as a union of non-redundantatoms of M . Theorem 13. i) Two atomizations of the same model have the same non-redundant atoms.ii) Any model has a unique atomization without redundant atoms.Proof. Let A and B be two atomizations of a model M without redundant atoms. Choose anon-redundant atom φ in B and consider the model [ A + { φ } ] spawned by φ and the atoms of A . It is clear that A + { φ } spawns the same model than A , otherwise there would be a duplethat is positive for [ A ] discriminated only by φ in [ A + { φ } ] and, hence, also discriminated by φ in [ B ], i.e. negative for [ B ], contradicting that A and B are atomizations of the same model.From theorem 9 either φ is a non-redundant atom in [ A ] or φ is redundant with [ A ]. Assume φ is redundant with [ A ]. There is a set E φ of atoms of A such that φ is a union of the atomsin E φ (see theorem 12). Choose an atom η in E φ and consider the model [ B + { η } ]. The samereasoning applies, so we should get that either η is in B or is redundant with atoms of B . Wecan substitute η with the atoms that make η redundant with B , and do the same for everyatom of E φ to form a set E ′ φ of atoms of B . It follows that φ is a union of the atoms in E ′ φ which implies that φ is redundant with B , against our assumptions. Therefore, φ cannot beredundant with A and then φ should be a non-redundant atom in A , which proves propositioni, i.e. that all the non-redundant atoms of B are in A and vice-versa. Finally, proposition iifollows from proposition i and theorem 9.If A and B are models or sets of atoms, we use A + B to represent the model [ A ∪ B ], amodel atomized by the atoms of A and B . Theorem 14. Let A and B be two sets of atoms that atomize models A and B over C . T h +0 ( A + B ) = T h +0 ( A ) ∩ T h +0 ( B ) and T h − ( A + B ) = T h − ( A ) ∪ T h − ( B ) . roof. Let r = ( a, b ) be a duple over C . From the third axiom, AS3, and theorem 2(iv) followsthat r ∈ T h − ( A ∪ B ) if and only if the discriminant dis A + B ( r ) is not empty. Operating on thedefinition of discriminant dis A + B ( r ) = L aA + B ( a ) − L aA + B ( b ) = L aA ( a ) ∪ L aB ( a ) − L aA ( b ) ∪ L aB ( b ). Dueto axiom AS5 sets A and B may have a non-empty intersection. However, whether an atom is inthe lower segment of terms a or b depends only on the atom, so L aA ( a ) ∩ L aB ( b ) ⊆ L aA ( a ) ∩ L aA ( b )and L aB ( a ) ∩ L aA ( b ) ⊆ L aB ( a ) ∩ L aB ( b ), and from here if follows that L aA ( a ) ∪ L aB ( a ) − L aA ( b ) ∪ L aB ( b ) =( L aA ( a ) − L aA ( b )) ∪ ( L aB ( a ) − L aB ( b )) and, finally, dis A + B ( r ) = dis A ( r ) ∪ dis B ( r ). Therefore, r is discriminated in A + B if and only if it is discriminated either in A or in B , and then T h − ( A + B ) = T h − ( A ) ∪ T h − ( B ). Taking the complementary sets we get T h +0 ( A + B ) = T h +0 ( A ) ∩ T h +0 ( B ) or, in other words, r is not discriminated in A + B if and only if r is notdiscriminated neither in A nor in B . Theorem 15. Let A and B be two sets of atoms that atomize models A and B , with model A freer or as free as B .i) The model A + B spawned by the atoms of A and the atoms of B is the same as the modelspawned by A alone.ii) The atoms of B are atoms of A or are redundant with the atoms of A .iii) A is freer than B if and only if the atoms of B are atoms of A or unions of atoms of A .Proof. (i) Since A is freer or as free as B all the negative duples of B are also negative in A .This means that a duple discriminated by an atom of B is also discriminated by some atom of A . In addition, each positive duple of A is also a positive duple of B and also a positive dupleof A + B . Therefore, a duple of A + B is positive if and only if is positive in A . Likewise, aduple is negative if and only if is negative in A and then models A + B and A are equal.(ii) follows from theorem 13 and the fact that A + B spawns the same model than A .(iii) Left to right. Assume A is freer than B . Consider the model A + B spawned by the atomsof A and the atoms of B . Proposition (i) tells us that if A is freer than B the model A + B isequal to A . Theorem 9 says that each atom φ of B is an atom of A or is redundant with theatoms of A . In other words, either φ is an atom of A or for each constant c such that φ < c there is at least one atom η < c in A such that φ is wider than η , i.e. the set U c ( φ ) contains theset U c ( η ). Since there is an η for each constant c ∈ U c ( φ ), if the set { η i : i = 1 , ..., n } makes φ redundant in A , then U c ( φ ) = ∪ i U c ( η i ) and φ is the union φ = ▽ i η i .Right to left. Assume now that the atoms of B are atoms of A or unions of atoms of A . If anatom φ of B is a union of atoms { η i : i = 1 , ..., n } of A then φ is redundant with A and anyduple discriminated by φ is discriminated by some of the atoms { η i : i = 1 , ..., n } . Therefore,any duple r discriminated by an atom of B is discriminated by at least one atom of A so, B | = r − implies A | = r − and A is freer or equal to B .18 efinition 3.2. Let r = ( r L , r R ) be a duple and M an atomized semilattice model, both overthe same set of constants. The full crossing of r in M , written (cid:3) r M , is the model atomized by( M − H ) ∪ ( H ▽ B ) where H = dis M ( r ), B = L aM ( r R ) and H ▽ B ≡ { λ ▽ ρ : ( λ ∈ H ) ∧ ( ρ ∈ B ) } .We define F C ( R + ) as the freest model that satisfies a set of positive duples R + . The nexttheorem gives a recipe to construct freest models. Theorem 16. Let M be an atomized model with or without redundant atoms and r a duplesuch that M | = r − . The full crossing of r in M is the freest model F C ( T h +0 ( M ) ∪ r + ) .Proof. Let H ⊆ M be the discriminant of r = ( r L , r R ), i.e. the set of atoms λ such that λ < r L ∧ λ < r R . Let B ⊆ M be the set of atoms of r R , i.e. the atoms ρ such that ρ < r R . The full crossing of ( r L , r R ) in model M is the model K = ( M − H ) ∪ ( H ▽ B ) where H ▽ B ≡ { λ ▽ ρ : ( λ ∈ H ) ∧ ( ρ ∈ B ) } is the set of all pairwise unions of an atom of H and anatom of B .Using the properties of theorem 11 (property iii) follows that the atoms λ ▽ ρ ∈ H ▽ B ⊆ K introduced by the full crossing operation satisfy λ ▽ ρ < r R because ρ < r R . Since the atoms inthe discriminant λ ∈ H = dis M ( r ) are no longer present in M − H and the atoms introducedin H ▽ B are all in the lower segment of r R then K | = r + .If a model N is atomized by a subset of the atoms of M then there is no duple r positive in M and negative in N . This should be clear, from theorem 14 since T h − ( M ) = T h − ( M + N ) = T h − ( M ) ∪ T h − ( N ) from which T h − ( N ) ⊆ T h − ( M ). In other words, the elimination of atomsfrom a model cannot cause any positive duple to become negative. Hence, the atoms in H eliminated by the full crossing operation cannot switch positive duples into negative. We haveto prove that the atoms introduced by the full crossing in H ▽ B do not switch positive duplesinto negative duples either. Assume M | = s + for some duple s = ( s L , s R ). Suppose that term s L acquires one of the new atoms λ ▽ ρ in its lower segment. From 11 (property iv) followsthat either λ < s L or ρ < s L . Because M | = ( s L ≤ s R ) and λ , ρ are atoms of M then( λ < s R ) ∨ ( ρ < s R ). Using now 11 (property iii) we get λ ▽ ρ < s R . Therefore, the atoms ofthe form λ ▽ ρ cannot switch a positive duple s + into s − .So far we know that K | = r + and ( M | = s + ) ⇒ ( K | = s + ), so the model resulting from fullcrossing satisfies K | = T h +0 ( M ) ∪ r + . To prove that K is the freest model of T h +0 ( M ) ∪ r + wehave to show that full crossing does not switch negative duples into positive either unless theyare logical consequences of T h +0 ( M ) ∪ r + .Consider a duple s = ( s L , s R ) such that K | = s + and M | = s − . The crossing of r has switched s from negative to positive. For this to occur a necessary condition is that the discriminant of s should disappear from K as a result of the crossing, in other words, dis M ( s ) ⊆ H = dis M ( r ),so the atoms of dis M ( s ) should all be atoms of r L . This implies s L ≤ s R ⊙ r L holds in M .19ince M | = s − there is at least one λ ∗ ∈ dis M ( s ). From 11, property iii, all the atoms of theform { λ ∗ ▽ ρ : ρ ∈ B } are in s L . If K | = s + then all the atoms λ ∗ ▽ ρ should also be atoms of s R in model K . Using 11 (property v) λ ∗ < s R and λ ∗ ▽ ρ < s R implies ρ < s R for each ρ ∈ B .Since B is the lower atomic segment of r R in M immediately follows that M | = ( r R ≤ s R ).Putting both conditions together: M | = (( s L ≤ s R ⊙ r L ) ∧ ( r R ≤ s R )) . Since ( s L ≤ s R ⊙ r L ) ∧ ( r R ≤ s R ) ∧ ( r L ≤ r R ) ⇒ s L ≤ s R s + is a logical consequence of the theory T h +0 ( M ) ∪ r + so any model of T h +0 ( M ) ∪ r + mustsatisfy s + . We have proved that any duple s + satisfied by K is satisfied by any model of T h +0 ( M ) ∪ r + and, because K | = T h +0 ( M ) ∪ r + then K = F C ( T h +0 ( M ) ∪ r + ).Finally, we should make sure that K satisfies the sixth axiom of atomized semilattices. Thisreduces to check that by removing the atoms in H from M we are not leaving some constant c with an empty lower atomic segment. Because for each λ < c we have at least one λ ▽ ρ < c in K , it follows that K satisfies the sixth axiom if M does.We use (cid:3) a ≤ b M or (cid:3) r M for a duple r = ( a, b ) to represent the model resulting from thefull crossing of r in M. Example 3.1. Suppose the model over the constants C = { a, b, c, d, e } : M = [ φ a , φ a,b , φ c,d,e , φ b,e , φ c , φ d ] . where we are using φ A for an atom with upper constant segment equal to the set A .Note that M | = ( b a ⊙ d ) as the duple ( b, a ⊙ d ) is discriminated by dis M ( b, a ⊙ d ) = { φ b,e } .The full crossing of ( b, a ⊙ d ) on M is equal to: (cid:3) b ≤ a ⊙ d M = [( M − φ b,e ) ∪ φ b,e ▽ { φ a , φ a,b , φ c,d,e , φ d } ] == [ { φ a , φ a,b , φ c,d,e , φ c , φ d } ∪ { φ a,b,e , φ a,b,e , φ b,c,d,e , φ b,d,e } ] == [ φ a , φ a,b , φ c,d,e , φ c , φ d , φ a,b,e , φ b,c,d,e , φ b,d,e ] . Since φ b,c,d,e = φ c,d,e ▽ φ b,d,e then φ b,c,d,e is redundant with (cid:3) b ≤ a ⊙ d M , and then: (cid:3) b ≤ a ⊙ d M = [ φ a , φ a,b , φ c,d,e , φ c , φ d , φ a,b,e , φ b,d,e ] , that is the freest model that satisfies (cid:3) b ≤ a ⊙ d M | = ( b ≤ a ⊙ d ) ∪ T h +0 ( M ) . heorem 17. The full crossing operation is commutative up to redundant atoms. The full-crossing of every duple in an ordered set of duples R produces the same model for every orderchosen.Proof. Theorem 16 shows that the result of full crossing the duples of R + in a model M isalways the freest model F C ( T h +0 ( M ) ∪ R + ) and, hence, is independent of the order of crossing.Because the result is the same model, from theorem 9, the resulting atomizations for differentorders of crossing can only differ in redundant atoms. Theorem 18. The freest model F C ( ∅ ) over a set C of constants has | C | non-redundant atoms,each with a single constant in its upper segment.Proof. The freest model is the term algebra, i.e. the model spawned by the terms modulusthe rules of the algebra, in this case the commutative, associative and idempotent laws. Forterms s and t , the term algebra satisfies that F C ( ∅ ) | = s ≤ t if and only if the componentconstants satisfy C ( s ) ⊆ C ( t ). Let M be the atomized model obtained by | C | different atoms,each with one constant in its upper segment. From the axiom of atomized models φ < a ⊙ b ⇔ ( φ < a ) ∨ ( φ < b ) applied to the component constants of s and t follows that for M (and forany semilattice), C ( s ) ⊆ C ( t ) implies s ≤ t . Conversely, assume M | = s ≤ t . Each atom φ in the lower atomic segment of s should be in the lower segment of some component constant c ∈ C ( s ) (this is proposition (ii) of theorem 2) and, since the atoms of M have only one constantin its upper segment then φ < t can occur only if c is also a component constant of t . Sinceeach constant of C has its own atom, every component constant of s should be a componentconstant of t otherwise there is an atom that discriminates ( s, t ) against our assumption. Sinceeach atom φ of M has an upper segment U c ( φ ) with a single constant, φ is non-redundant. Itfollows that M is an atomized semilattice model without redundant atoms that satisfies thesame positive an negative duples that F C ( ∅ ) and, hence, M is equal to F C ( ∅ ).We can easily show that the model M of theorem18, i.e. the model spawned by | C | differentatoms each with one constant in its upper segment, is freer than any atomized model. Let N be any atomized model and let r be any duple such that N | = r − . Each atom η of N is theunion of the atoms of M corresponding to each constant in the set U c ( η ) and this proves that η is redundant with M . Since all the atoms of N are redundant with M , using theorem 15proposition (iii) we conclude that M is freer or as free as N and, therefore, M | = r − and M isthe freest model. From theorem 9 any atomization of F C ( ∅ ) contains the | C | atoms of M andonly can differ from M in a set of redundant atoms.21 heorem 19. Any model M with a finite set C of constants can be atomized.Proof. Since the theory T h +0 ( M ) for any model M over a finite set of constants is a finite a set, M can be atomized by starting with the set of | C | atoms that provides an atomization for thefreest model F C ( ∅ ) (see theorem 18), each atom contained in a single constant, and then byperforming a finite sequence of full crossing operations for each duple in T h +0 ( M ). As a result weobtain an atomization of model M which follows from theorem 16 and M = F C ( T h +0 ( M )).The reader may recognize theorem 19 as a consequence of the Stone’s theorem [3]. Intheorem 37 we will make the connection explicit. Theorem 20. The models resulting from the full crossing of a duple r in different atomizationsof a model M can only differ in redundant atoms.Proof. The proof of theorem 16 needs no assumption regarding redundant atoms. With orwithout redundant atoms in the atomization of a model M the result of full crossing a duple r in M is the freest model F C ( T h +0 ( M ) ∪ r + ). Theorem 9 assures us that if the resulting modelsare the same then the atomizations can only differ in redundant atoms. Theorem 21. Let M be an atomized semilattice. If M | = r − then M is freer than (cid:3) r M . If M | = r + then M = (cid:3) r M .Proof. Theorem 16 states that (cid:3) r M is the freest model F C ( T h +0 ( M ) ∪ { r + } ). Since (cid:3) r M | = T h +0 ( M ) ∪ { r + } then T h +0 ( M ) ⊂ T h +0 ( (cid:3) r M ). Since (cid:3) r M and M are generated by the sameconstants, T h +0 ( M ) ⊂ T h +0 ( (cid:3) r M ) is equivalent to T h − ( (cid:3) r M ) ⊂ T h − ( M ), and it follows that M is freer than (cid:3) r M .If M | = r + the discriminant of r in M is empty and full crossing leaves M unaltered. Alterna-tively, (cid:3) r M = F C ( T h +0 ( M ) ∪ { r + } ) = F C ( T h +0 ( M )) = M .22 Join models, restrictions, subalgebras and products We use M | Q or [ | Q ] M to refer to the “restriction of M to Q ”, a model defined as the subalgebraof M generated by Q . Alternatively we can define M | Q as the smallest subalgebra of M thatcontains the constants Q . Definition 4.1. Let Q be a subset of the constants Q ⊂ C . The “restriction of φ to Q ”, is theatom φ | Q with upper segment: U c ( φ | Q ) = U c ( φ ) ∩ Q . The restriction of φ to Q does not existif U c ( φ ) ∩ Q = ∅ . Theorem 22. Let M be an atomized semilattice model over a set of constants C and Q be asubset of C . Let M | Q be the subalgebra of M generated by Q . Then M | Q is the model spawnedby the restriction to Q of the atoms of M . The restriction of the atoms of a model M to asubset Q of the constants produces the same model M | Q for every atomization of M .Proof. Let r ≡ ( a, b ) be a duple and a and b terms over Q . Let φ be an atom of M and φ | Q itsrestriction to Q defined by the upper constant segment U c ( φ | Q ) = U c ( φ ) ∩ Q . Let C ( a ) be thecomponent constants of a . By definition, ( φ | Q < a ) holds when ( U c ( φ ) ∩ Q ) ∩ C ( a ) = ∅ and,since a is a term over Q , then U c ( φ ) ∩ C ( a ) = ∅ . It follows that ( φ | Q < a ) if and only if ( φ < a ).Hence, if r is discriminated by an atom of M then r is discriminated by the restriction to Q ofthe same atom. Additionally, if r is discriminated by no atom in M (i.e. M | = a ≤ b ) then norestriction to Q of any atom of M can discriminate r . Therefore, the model spawned by therestriction of the atoms is the subalgebra M | Q regardless of the atomization chosen for M .We have seen that the restriction of the atoms of M to Q spawns the well-defined semilattice M | Q . Theorem 13 tells us that the non-redundant atoms of a model are unique, so each non-redundant atom of M | Q has to be equal to the restriction to Q of at least one atom of M .In addition, we could choose for M an atomization containing only non-redundant atoms in M which implies that each non-redundant atom of M | Q is the restriction of at least one non-redundant atom in M . We provide here another more direct proof that the reader may findilluminating: Theorem 23. Let M be an atomized semilattice model over C and let Q ⊂ C . For each non-redundant atom α of M | Q there is at least one non-redundant atom of M that restricted to Q is equal to α . roof. Theorem 22 says that each α in M | Q is the restriction of some atom of φ ∈ M . i.e. α = φ | Q . Either φ is a non-redundant atom of M or φ is redundant with M and then φ = ▽ ni ϕ i where { ϕ , ϕ , ..., ϕ n } are all non-redundant atoms in M . From U c ( φ ) = ∪ i U c ( ϕ i ) we get U c ( φ ) ∩ Q = ∪ i ( U c ( ϕ i ) ∩ Q ) so we can write φ | Q = ▽ mk ϕ | Qk with m ≤ n where the summationwith index k runs along the ϕ | Q for which U c ( ϕ k ) ∩ Q = ∅ , in other words, it runs along the ϕ | Q that exist. Again, theorem 22 tells us that ϕ | Qk ∈ M | Q . If α is non-redundant in M | Q ,since α = φ | Q = ▽ mk ϕ | Qk , then α can be equal to a union of atoms of M | Q only if at leastone of the atoms in the union is equal to α (otherwise α would be redundant with M | Q ), i.e. α = φ | Q = ϕ | Qk for at least one value of k . It follows that if α is non-redundant in M | Q then α is equal to the restriction of at least one non-redundant atom in M . Theorem 24. Let M be an atomized semilattice model over C and let Q ⊂ C . There is ahomomorphism from M onto M | Q that maps each term over C to its restriction to Q if andonly if ⊖ Q is a non-redundant atom of M | Q .Proof. Assume ⊖ Q is a non-redundant atom of M | Q . Theorem 6 says that there is at least oneconstant k ∈ Q such that M | Q | = ∀ x ( k ≤ x ). Let h map each constant of C − Q to k and eachconstant in Q to the constant with the same name in M | Q . Each term t over C is either a termover Q or a term over C − Q or an idempotent summation of a term r over Q and a term s over C − Q . Since h ( r ⊙ s ) = h ( r ) ⊙ h ( s ) = r ⊙ k = r , then h is a homomorphism from M ontoits subalgebra M | Q that maps each term over C to its restriction to Q .On the the other direction, assume that there is a homomorphism h from M onto its subalgebra M | Q that maps each term over C to its restriction to Q . Choose a term s over C − Q . For everyterm r over Q we have h ( r ) ⊙ h ( s ) = h ( r ⊙ s ) and, since h maps r ⊙ s to its restriction to Q then h ( r ⊙ s ) = h ( r ) = r . It follows that h ( s ) behaves as neutral element, i.e. h ( s ) < x for every x in M | Q and, again from theorem 6, a neutral element in M | Q implies ⊖ Q is a non-redundantatom of M | Q .Note that it is is always possible to add a new constant to C and Q that behaves as aneutral element and then build a homomorphism from M onto M | Q that maps each term over C to its restriction to Q . Theorem 25. Let r be a duple over Q ⊂ C and M a model over C :i) dis [ | Q ] M ( r ) = [ | Q ] dis M ( r ) ,ii) Restriction and crossing commute: (cid:3) r [ | Q ] M = [ | Q ] (cid:3) r M . roof. (i) Let r = ( r L , r R ). The discriminant of r in [ | Q ] M is: dis [ | Q ] M ( r ) = { φ : ( φ ∈ [ | Q ] M ) ∧ ( φ < r L ) ∧ ( φ < r R ) } = { φ | Q : ( φ ∈ M ) ∧ ( φ | Q < r L ) ∧ ( φ | Q < r R ) } , where we have used theorem 23. Since r is a duple over Q then: { φ | Q : ( φ ∈ M ) ∧ ( φ | Q < r L ) ∧ ( φ | Q < r R ) } = { φ | Q : ( φ ∈ M ) ∧ ( φ < r L ) ∧ ( φ < r R ) } = | Q ] dis M ( r ) . (ii) From the definition of full crossing: (cid:3) r [ | Q ] M = ([ | Q ] M − dis [ | Q ] M ( r )) + dis [ | Q ] M ( r ) ▽ L a [ | Q ] M ( r R ) == ([ | Q ] M − [ | Q ] dis M ( r )) + ([ | Q ] dis M ( r )) ▽ ([ | Q ] L aM ( r R )) . where we have used proposition (i) and taken into account that r R is a term over Q .Consider the sets of atoms A = M − dis M ( r ) and B = [ | Q ] M − [ | Q ] dis M ( r ). We are going toshow now that [ | Q ] A = B . Suppose there is an atom α ∈ [ | Q ] A such that α B . Because α ∈ [ | Q ] A implies α ∈ [ | Q ] M then α B requires α ∈ [ | Q ] dis M ( r ). Since r is a duple over Q , for every α ∈ [ | Q ] dis M ( r ) we have that ∀ φ (( α = φ | Q ) ⇒ ( φ ∈ dis M ( r ))) which implies ∀ φ (( α = φ | Q ) ⇒ ( φ A )). Since theorem 23 says that such φ should exist in M , then α [ | Q ] A , contradicting our assumption. Hence: [ | Q ] A ⊆ B .Suppose now there is an atom β ∈ B such that β [ | Q ] A . Like before, β ∈ B implies β ∈ [ | Q ] M and then β [ | Q ] A requires ∀ φ (( β = φ | Q ) ⇒ ( φ ∈ dis M ( r ))), which in turn, implies β ∈ [ | Q ] dis M ( r ) and then β B ; a contradiction. It follows B ⊆ [ | Q ] A and considering theinclusion in the other direction [ | Q ] A = B .Since every atom in the discriminant dis M ( r ) and every atom in L aM ( r R ) have a restriction to Q then ([ | Q ] dis M ( r )) ▽ ([ | Q ] L aM ( r R )) = [ | Q ]( dis M ( r ) ▽ L aM ( r R )). Substituting above and alsoreplacing B by [ | Q ] A we get ([ | Q ] M − [ | Q ] dis M ( r )) + ([ | Q ] dis M ( r )) ▽ ([ | Q ] L aM ( r R )) = [ | Q ] (cid:3) r M ,which proves the theorem. Theorem 26. Let C be a set of constants and Q ⊆ C . Let M be a semilattice model over C and N a semilattice model over Q . Let r be a duple over Q . If M is as free or freer than N then (cid:3) r M is as free or freer than (cid:3) r N .Proof. M is freer or as free as N , therefore T h − ( N ) ⊆ T h − ( M ). Assume first that Q = C .When Q = C , it is also true that T h +0 ( M ) ⊆ T h +0 ( N ). Suppose there is a duple s ∈ T h − ( N )for which (cid:3) r N | = s − and (cid:3) r M | = s + . Consider the union model (cid:3) r N + (cid:3) r M spawned by theatoms of both models. From theorem 14 follows that (cid:3) r N + (cid:3) r M | = T h +0 ( M ) ∪ { r + } ∪ { s − } .Theorem 16 says that (cid:3) r M is the freest model that satisfies T h +0 ( M ) ∪ { r + } , i.e. (cid:3) r M = F C ( T h +0 ( M ) ∪ { r + } ) and we have assumed (cid:3) r M | = s + . However, model (cid:3) r N + (cid:3) r M is amodel of T h +0 ( M ) ∪ { r + } that do not satisfy s + which contradicts that the freest model satis-fies s + . Therefore, such s does not exist and then T h − ( (cid:3) r N ) ⊆ T h − ( (cid:3) r M ).25onsider now that Q ⊂ C . All the duples in T h − ( N ) are duples over Q . Since every pos-itive or negative duple over Q satisfied by M is also satisfied by its subalgebra [ | Q ] M then T h − ( N ) ⊆ T h − ([ | Q ] M ) = T h − ( M ) ∩ ( F Q ( ∅ ) × F Q ( ∅ )). Since N and [ | Q ] M are modelsover the same constants, Q , we have just shown that T h − ( N ) ⊆ T h − ([ | Q ] M ) implies that T h − ( (cid:3) r N ) ⊆ T h − ( (cid:3) r [ | Q ] M ) and, using theorem 25 which says that crossing and restrictioncommute, T h − ( (cid:3) r N ) ⊆ T h − ([ | Q ] (cid:3) r M ) ⊆ T h − ( (cid:3) r M ) and then (cid:3) r M is as free or freer than (cid:3) r N . Theorem 27. Let C be a set of constants and Q ⊆ C . Let M be a semilattice model over C and N a semilattice model over Q . The following statements are equivalent:i) M is as free or freer than N ,ii) There is a set of duples { r , r , ..., r v } over C such that N + [ ⊖ c ] = (cid:3) r (cid:3) r ... (cid:3) r v M ,iii) There is a set of duples { r , r , ..., r u } over Q such that N = [ | Q ] (cid:3) r (cid:3) r ... (cid:3) r u M ,iv) N is a subset model of M , written N ⊆ M , i.e. the atoms of N are all atoms of M orredundant with M .Proof. ( i ) ⇒ ( ii ) Proposition (i) assumes T h − ( N ) ⊆ T h − ( M ), and since ⊖ c discriminates noduple T h − ( N + [ ⊖ c ]) = T h − ( N ) and then T h − ( N + [ ⊖ c ]) ⊆ T h − ( M ). Because N + [ ⊖ c ]is a model over C , it holds that T h +0 ( M ) ⊆ T h +0 ( N + [ ⊖ c ]). Consider the set of duples R = { r , r , ..., r v } = T h − ( M ) − T h − ( N + [ ⊖ c ]) = T h +0 ( N + [ ⊖ c ]) − T h +0 ( M ). It follows that (cid:3) r (cid:3) r ... (cid:3) r v M produces the freest model of T h +0 ( M ) ∪ R + = T h +0 ( N + [ ⊖ c ]), a model that isequal N + [ ⊖ c ].( ii ) ⇒ ( i ) According to theorem 21, full crossing either leaves the model unaltered or producesstrictly less free models. Therefore, M is as free or freer than N + [ ⊖ c ], in other words, T h − ( N + [ ⊖ c ]) ⊆ T h − ( M ) and, since T h − ( N + [ ⊖ c ]) = T h − ( N ), then T h − ( N ) ⊆ T h − ( M ) and M is as free or freer than N .( i ) ⇒ ( iii ) Proposition (i) assumes T h − ( N ) ⊆ T h − ( M ). Since T h − ( N ) is a set of duplesover Q then T h − ( N ) ⊆ T h − ( M ) ∩ ( F Q ( ∅ ) × F Q ( ∅ )) = T h − ([ | Q ] M ). Consider the set ofduples R = { r , r , ..., r u } = T h − ([ | Q ] M ) − T h − ( N ) = T h +0 ( N ) − T h +0 ([ | Q ] M ). Note that N | = r + for every r ∈ R . Since both are models over Q then T h +0 ([ | Q ] M ) ⊆ T h +0 ( N ) andit follows that (cid:3) r (cid:3) r ... (cid:3) r u [ | Q ] M produces the freest model of T h +0 ([ | Q ] M ) ∪ R + = T h +0 ( N ),a model that is equal to N . Theorem 25 says that full crossing and restriction commute so[ | Q ] (cid:3) r (cid:3) r ... (cid:3) r u M = N .( iii ) ⇒ ( i ) Theorem 25 implies N = [ | Q ] (cid:3) r (cid:3) r ... (cid:3) r u M = (cid:3) r (cid:3) r ... (cid:3) r u [ | Q ] M and, then,theorem 21 says that [ | Q ] M is as free or freer than N , i.e, T h − ( N ) ⊆ T h − ([ | Q ] M ). Since T h − ( N ) is a set of duples over Q and every positive or negative duple over Q satisfied by M is also satisfied by its subalgebra [ | Q ] M then T h − ( N ) ⊆ T h − ([ | Q ] M ) = T h − ( M ) ∩ ( F Q ( ∅ ) × F Q ( ∅ )). It follows T h − ( N ) ⊆ T h − ( M ), and M is as free or freer than N .26 ii ) ⇒ ( iv ) Since N + [ ⊖ c ] = (cid:3) r (cid:3) r ... (cid:3) r v M then the atoms of N are the result of successivecrossing operations and then they are all either atoms of M or union of atoms of M , hence,redundant with M .( iv ) ⇒ ( i ) Theorem 9 tells us that atoms redundant with M only discriminate duples that arediscriminated by atoms of M , therefore T h − ( N ) ⊆ T h − ( M ). Theorem 28. i) Let M and N be two semilattice models over C . If M is as free or freer than N then there is a homomorphism from M onto N .ii) Let C be a set of constants and Q ⊆ C . Let M be a semilattice model over C and N asemilattice model over Q . If M is as free or freer than N there is a homomorphism from M | Q onto N .Proof. (i) M is as free or freer than N , so T h − ( N ) ⊆ T h − ( M ) and since M and N are bothgenerated by the same constants, T h +0 ( M ) ⊆ T h +0 ( N ). Let t and s be terms over C . Suppose N | = ( t ≤ s ) and M | = ( t s ), this is equivalent to N | = ( t = s ⊙ t ) and M | = ( t = s ⊙ t ), i.e.each difference between T h +0 ( M ) and T h +0 ( N ) reduces to some equality satisfied by N and notsatisfied by M . Consider the partition of the universe of M that puts elements a and b of M inthe same class if and only if N | = ( a = b ). Since T h +0 ( M ) is a subset of T h +0 ( N ) the partition isisomorphic to N and a map from each element of M to its partition class is a homomorphismfrom M onto N .(ii) In the proof of theorem 27 we argued that if M is as free or freer than N then M | Q is also asfree or freer than N with both, M | Q and N , over Q so from part (i) there is a homomorphismfrom M | Q onto N .A congruence [3, 4] θ of M is an equivalence relation of M that commutes with theoperations of the algebra. In the particular case of semilattices is an equivalence relation thatcommutes with the idempotent operator. θ is a congruence if: a θb ∧ a θb ⇒ ( a ⊙ a ) θ ( b ⊙ b ) . Let a and b be two elements of M . Congruences can be built from principal congruences Θ( a, b )defined as the smallest congruence with a and b in the same equivalence class. It is well known[3] that any congruence θ = ∪{ Θ( a, b ) : ( a, b ) ∈ θ } = ∨{ Θ( a, b ) : ( a, b ) ∈ θ } where ∪ is theunion of equivalence classes and ∨ is the join operator in the lattice of equivalence classes.The next theorem shows that any congruence on a finite semilattice can be constructed asa sequence of full crossings. 27 heorem 29. Let Θ( a, b ) be a principal congruence of an atomized semilattice M over C with a, b elements of M :i) (cid:3) a ≤ b M ≈ M/ Θ( b, b ⊙ a ) ,ii) M/ Θ( a, b ) ≈ (cid:3) a ≤ b (cid:3) b ≤ a M .Proof. Theorem 16 says that (cid:3) a ≤ b M is the freest model F C ( T h +0 ( M ) ∪ { ( a ≤ b ) } ). Since M isfreer than (cid:3) a ≤ b M theorem 28 says that we can build a homomorphism h from M to N = (cid:3) a ≤ b M mapping h : ν M ( t ) ν N ( t ) where ν M and ν N are the natural homomorphisms and t is a termover C . It follows from the Homomorphism Theorem [3] of Universal Algebra that N is thequotient algebra (cid:3) a ≤ b M = M/ker ( h ) where ker ( h ) is the kernel of the homomorphism.(i) It is well known [3] that ker ( h ) is also a congruence on M and, since (cid:3) a ≤ b M | = ( a ≤ b )and ( a ≤ b ) ⇔ ( b = b ⊙ a ) then ( b, b ⊙ a ) ∈ ker ( h ). The principal congruence is the smallestcongruence Θ( b, b ⊙ a ) that has ( b, b ⊙ a ), then Θ( b, b ⊙ a ) ⊆ ker ( h ). On the other hand, M/ Θ( b, b ⊙ a ) | = T h +0 ( M ) ∪ { ( b = b ⊙ a ) } . Suppose Θ( b, b ⊙ a ) is strictly smaller than ker ( h ),then M/ Θ( b, b ⊙ a ) is strictly freer than (cid:3) a
Let M and N be two atomized semilattices over the constants C M and C N respectively, with C M ∩ C N = ∅ . Rename the constants in the set { c , c , .., c h } = C M ∩ C N toobtain disjoint sets C M and C ′ N = [ c ′ c ... c ′ h c h ] C N and let the renamed model N ′ = [ c ′ c ... c ′ h c h ] N .i) M ⊕ N satisfies M ⊕ N ≈ (cid:3) E ( M + N ′ ) where (cid:3) E = (cid:3) Ec (cid:3) Ec ... (cid:3) Ec h and (cid:3) Ec = (cid:3) c ≤ c ′ (cid:3) c ′ ≤ c .ii) M ⊕ N = [ | ( C M ∪ C N )] (cid:3) E ( M + N ′ ) .iii) The atoms of M ⊕ N are atoms of M + N or redundant with M + N .iv) M + N is freer or as free than M ⊕ N .v) There is a homomorphism from M + N onto M ⊕ N .vi) M ⊕ N = (cid:3) r +1 (cid:3) r +2 .... (cid:3) r + k ( M + N ) for some duples { r +1 , r +2 , .., r + k } over C M ∪ C N .Proof. (i) It follows directly from the definition of the join model and theorem 29 to transformcongruences into full crossing operations: M ⊕ N = [ c c ′ ... c h c ′ h ] (cid:3) E ( M + N ′ ), and then M ⊕ N isa rename of (cid:3) E ( M + N ′ ), so both models are isomorphic.(ii) (cid:3) E equates each pair of constants c i = c ′ i for 1 ≤ i ≤ h . Since (cid:3) E ( M + N ′ ) | = ∀ i ( c i = c ′ i )the atoms in the lower segments of c i and c ′ i are the same, and then every atom with c i inits upper constant segment also has c ′ i and vice versa. Therefore, (cid:3) E ( M + N ′ ) is the model[ c c ′ c ... c h c ′ h c h ]( M ⊕ N ). Let A = [ c c ′ c ... c h c ′ h c h ]( M ⊕ N ) = (cid:3) E ( M + N ′ ). From theorem 22, therestriction to C M ∪ C N of A recovers M ⊕ N (as the restriction simply acts by removing theprimed constants from the upper segments of the atoms) and we can write this as A | ( C M ∪ C N ) = M ⊕ N or in the form of an operator, [ | ( C M ∪ C N )] A = M ⊕ N .(iii) Since A = (cid:3) E ( M + N ′ ) is the result of a crossing operation over M + N ′ then M + N ′ isfreer than A . This can be written as A ⊂ M + N ′ , where the inclusion signifies that the atomsof A are atoms of M + N ′ or redundant with M + N ′ . Let φ be an atom of A . We argued abovethat if c i ∈ C M ∩ C N then either both constants c i , c ′ i ∈ U c ( φ ), or c i U c ( φ ) and c ′ i U c ( φ ).This implies that if φ is a union of an atom in M and an atom in N ′ then φ | ( C M ∪ C N ) is a unionof an atom in M and an atom in N . We have in A two kinds of atoms. Atoms that are unionsof atoms of M and N that satisfy φ | ( C M ∪ C N ) ∈ M + N and atoms that are either in M or in N that contain no constant in the set C M ∩ C N and satisfy φ | ( C M ∪ C N ) = φ ∈ M + N . It followsthat every atom of A | ( C M ∪ C N ) = M ⊕ N is either an atom of M + N or a union of atoms of M + N . We can also write this as M ⊕ N ⊂ M + N .(iv) Proposition iii implies that M ⊕ N is a subset model of M + N so, using theorem 27(iv)we get that M + N is freer or as free as M ⊕ N .(v) Directly from part (iv) and theorem 28. 30vi). From part iv and the fact that M + N and M ⊕ N are both models over the same constants C M ∪ C N , follows that T h +0 ( M + N ) ⊆ T h +0 ( M ⊕ N ) so there is some set R + of duples over C M ∪ C N , perhaps empty, such that T h +0 ( M ⊕ N ) = F C M ∪ C N ( T h +0 ( M + N ) ∪ R + ), for example,the set R + = T h +0 ( M ⊕ N ) − T h +0 ( M + N ). Let’s enumerate the set R + = { r +1 , r +2 , .., r + k } . Wecan use theorem 16 to build the freest model that satisfies T h +0 ( M + N ) ∪ R + as a series offull-crossings M ⊕ N = (cid:3) r +1 (cid:3) r +2 .... (cid:3) r + k ( M + N ).We have shown that M + N is freer but not equal to M ⊕ N . How do they look? Example 4.1. Let M be a semilattice model of an algebra with constants C M = { a, b, c } and N a model with constants C N = { c, d, e } . Assume that M = [ φ c , φ a,b,c ] and N = [ φ c,d,e ] , wherewe are using the same notation for atoms than in the example 3.1.It easily follows that: M | = ( a = b < c ) N | = ( c = d = e ) . The union model M + N is equal [ φ c , φ a,b,c , φ c,d,e ] and is a model that satisfies: M + N | = ( a = b < c > d = e ) . The join model M ⊕ N can be obtained by calculating [ |{ a, b, c, d, e } ] (cid:3) c ≤ c ′ (cid:3) c ′ ≤ c ( M + [ c ′ c ] N ) .Step by step: M + [ c ′ c ] N = [ φ c , φ a,b,c , φ c ′ ,d,e ] and then (cid:3) c ′ ≤ c [ φ c , φ a,b,c , φ c ′ ,d,e ] = [ φ c,c ′ ,d,e , φ a,b,c,c ′ ,d,e ] and (cid:3) c ≤ c ′ [ φ c,c ′ ,d,e , φ a,b,c,c ′ ,d,e ] = [ φ c,c ′ ,d,e , φ a,b,c,c ′ ,d,e ] . Finally, the join model is given by the restric-tion [ φ c,c ′ ,d,e , φ a,b,c,c ′ ,d,e ] |{ a,b,c,d,e } , which yields M ⊕ N = [ φ c,d,e , φ a,b,c,d,e ] , a model that satisfies: M ⊕ N | = ( a = b < c = d = e ) , and contains embeddings of both M and N . It is also clear that M + N is freer than M ⊕ N and, in fact, (see theorem 31 part vi) it can be obtained as: M ⊕ N = (cid:3) c ≤ d ( M + N ) . In the example above we saw that both M and N could be embedded in M ⊕ N but notin M + N . The next theorem explains when embeddings in the join model are possible. Theorem 32. Let M and N be two semilattice models over the constants C M and C N respec-tively.i) M ⊕ N = F C M ∪ C N ( T h +0 ( M ) ∪ T h +0 ( N )) .ii) Either T h + ( M | C N ) T h + ( N ) or there is an embedding from N into M ⊕ N with image ( M ⊕ N ) | C N = N .iii) Either there is a duple r over C N such that M | = r + and N | = r − or there is an embeddingfrom N into M ⊕ N with image ( M ⊕ N ) | C N = N .iv) If C M ∩ C N = ∅ there are embeddings from M and from N into M ⊕ N . roof. (i) From theorem 31ii we know M ⊕ N = [ | ( C M ∪ C N )] (cid:3) E ( M + N ′ ) where N ′ =[ c ′ c ... c ′ h c h ] N . Using theorem 16 we can write (cid:3) E ( M + N ′ ) = F C M ∪ C N ( T h +0 ( M + N ′ ) ∪ E ) with E = ∪ ∀ i ∈ C M ∩ C N { ( c i ≤ c ′ i ) ∧ ( c ′ i ≤ c i ) } . Since the constants of M and N ′ are disjoint, it isnot difficult to see that T h +0 ( M + N ′ ) ⇔ T h +0 ( M ) ∪ T h +0 ( N ′ ). It follows that (cid:3) E ( M + N ′ ) = F C M ∪ C N ( T h +0 ( M ) ∪ T h +0 ( N ′ ) ∪ E ). It is also clear that T h +0 ( N ′ ) ∪ E ⇔ T h +0 ( N ) ∪ E and then (cid:3) E ( M + N ′ ) = F C M ∪ C N ( T h +0 ( M ) ∪ T h +0 ( N ) ∪ E ), where primed constants c ′ i appear only in E and its restriction to C M ∪ C N is equal to F C M ∪ C N ( T h +0 ( M ) ∪ T h +0 ( N )).(ii) We can construct M ⊕ N = F C M ∪ C N ( T h +0 ( M ) ∪ T h +0 ( N )) with a series of full crossings (cid:3) T h +0 ( N ) F C M ∪ C N ( T h +0 ( M )) where (cid:3) T h +0 ( N ) is a sequence with a full crossing for each duple in T h +0 ( N ). Consider the restriction: ( M ⊕ N ) | C N = [ | C N ] (cid:3) T h +0 ( N ) F C M ∪ C N ( T h +0 ( M )). Since eachduple in T h +0 ( N ) is over the constants C N , restriction and full crossing commute (see theorem25) so we can write ( M ⊕ N ) | C N = (cid:3) T h +0 ( N ) [ | C N ] F C M ∪ C N ( T h +0 ( M )). Keeping theorem 18 inmind it is easy to see that F C M ∪ C N ( T h +0 ( M )) = M + F C N − C M ( ∅ ). Substituting above, we get( M ⊕ N ) | C N = (cid:3) T h +0 ( N ) [ | C N ] ( M + F C N − C M ( ∅ )) = (cid:3) T h +0 ( N ) ( M | C N + F C N − C M ( ∅ )). Consideringthat M | C N + F C N − C M ( ∅ ) = (cid:3) T h + ( M | CN ) F C N ( ∅ ) follows ( M ⊕ N ) | C N = (cid:3) T h +0 ( N ) (cid:3) T h + ( M | CN ) F C N ( ∅ ).Suppose T h + ( M | C N ) ⊆ T h + ( N ). It is clear, then, that (cid:3) T h +0 ( N ) (cid:3) T h + ( M | CN ) = (cid:3) T h +0 ( N ) and( M ⊕ N ) | C N = (cid:3) T h +0 ( N ) F C N ( ∅ ) = N . Since ( M ⊕ N ) | C N is a subalgebra of M ⊕ N there is anembedding form N into M ⊕ N .(iii) T h + ( M | C N ) T h + ( N ) occurs if and only if there is a duple r over C N such that M | C N | = r + and N | = r − and it is a consequence of theorem 22 that M | C N | = r + if and only if M | = r + , forany duple r over C N .(iv) if C M ∩ C N = ∅ there is no duple r satisfying the conditions of proposition (iii), so thereshould be an embedding from N into M ⊕ N (and another embedding from M into M ⊕ N ).Theorem 32 provides an alternative definition for the join model: M ⊕ N = F C M ∪ C N ( T h +0 ( M ) ∪ T h +0 ( N )) . We know how to do a restriction M | Q , which is a subalgebra generated by a subset Q ⊂ C of the constants. Consider the more general problem of calculating the subalgebra S of a model M over C generated by a set of e elements represented by terms { t , t , .., t e } over C . Let G be a set with e new constants. We can first extend the model M to a model M ⊕ F G over C ∪ G , and then equate each of the e terms to one constant in G using a principal congruenceas follows: S ≈ [ | G ] ( M ⊕ F G ( ∅ )) /θ θ = ∨ i Θ( t i , g i ) . where G = { g , g , .., g e } are constants. Once we calculate the quotient model ( M ⊕ F G ( ∅ )) /θ we do a restriction to the subset G of the constants to obtain a model over G that is isomorphicto the subalgebra generated by the terms { t , t , .., t e } .32 heorem 33. Let M be an atomized semilattice over C = { c , c , .., c m } and { t , t , .., t e } aset of terms of F C ( ∅ ) . Let G = { g , g , .., g e } be a set of constants. The subalgebra S of M generated by { t , t , .., t e } and represented using the constants in G is:i) S = [ | G ]Π ei =1 (cid:3) g i ≤ t i (cid:3) t i ≤ g i ( M + F G ( ∅ )) .ii) Let G k = { g k , g k , .., g ke ( k ) } ⊆ G such that ( g i ∈ G k ) ⇔ ( c k ∈ C ( t i )) . Then S is the renamedmodel: S = [ g g ...g e (1) c g g ...g e (2) c ... g m g m ...g me ( m ) c m ] M .iii) The number of non-redundant atoms of S is equal or smaller than the number of non-redundant atoms of M .Proof. (i) We argued that S ≈ [ | G ] ( M ⊕ F G ( ∅ )) /θ , where the congruence θ = ∨ i Θ( t i , g i ).Since C ∩ G = ∅ , theorem 30 says that we can build M ⊕ F G simply as M + F G , the unionof two sets of atoms. Since θ is a join of congruences we can write, for any model A , that A/θ = ( A/ Θ( t , g )) / Θ( t , g ) .../ Θ( t e , g e ) and theorem 29 allows us to build the congruencestep by step, in any order, using full crossing operations, as A/θ = Π ei =1 (cid:3) g i ≤ t i (cid:3) t i ≤ g i A .(ii) Theorem 18 says that F G ( ∅ ) can be atomized with e non-redundant atoms φ g , φ g , ..., φ g e each with a single constant g i in its upper segment. Since C ∩ G = ∅ the crossing (cid:3) t i ≤ g i ( M + F G ( ∅ )) simply adds g i to the upper segment of every atom φ < t i of M . The next crossing (cid:3) g i ≤ t i acts by removing φ g i from the resulting atomization. Since φ < t i if and only if thereis a constant c k ∈ C ( t i ) such that φ < c k the atom φ gains g i if and only if there is a c k suchthat φ < c k ∈ C ( t i ) or, in other words if U c ( φ ) ∩ C ( t i ) = ∅ . The result is that after computingΠ ei =1 (cid:3) g i ≤ t i (cid:3) t i ≤ g i ( M + F G ( ∅ )) all the atoms of F G ( ∅ ) are gone and every atom φ of M has gained g i if and only if U c ( φ ) ∩ C ( t i ) = ∅ . After calculating the full crossings, obtaining the restrictionto G can be done by simply removing all the original c constants from the upper segments ofthe atoms, eliminating every atom that has no constants of G in its upper segment. At theend, each atom of φ of M has changed to [ g g ...g e (1) c g g ...g e (2) c ... g m g m ...g me ( m ) c m ] φ and the propositionfollows from theorem 22. Note that some G k may be empty; in this case ∅ c k simply removes c k from the upper segment of the atoms.(iii) Take an atomization for M with just non-redundant atoms and calculate S using proposi-tion ii. The atoms of S are renamed atoms of M , hence, S cannot have more non-redundantatoms than M . Example 4.2. Consider the subalgebra of F c ,c ,c ( ∅ ) generated by the terms t = c , t = c , t = c ⊙ c and t = c ⊙ c . It follows: F c ,c ,c ( ∅ ) | = ( t < t ) ∧ ( t < t ) ∧ ( t ⊙ t = t ⊙ t ) . Applying the theorem 33(ii) we can calculate the subalgebra as: S = [ g g c g g c g g c ][ φ c , φ c , φ c ] = [ φ g g , φ g g , φ g g ] , model over G = { g , g , g , g } that satisfies: S | = ( g < g ) ∧ ( g < g ) ∧ ( g ⊙ g = g ⊙ g ) . Same result should be obtained from theorem 33(i): S = [ | G ]Π ei =1 (cid:3) g i ≤ t i (cid:3) t i ≤ g i [ φ c , φ c , φ c , φ g , φ g , φ g , φ g ] == [ | G ] (cid:3) g ≤ c ⊙ c (cid:3) c ⊙ c ≤ g (cid:3) g ≤ c ⊙ c (cid:3) c ⊙ c ≤ g (cid:3) g ≤ c (cid:3) c ≤ g (cid:3) g ≤ c (cid:3) c ≤ g [ φ c , φ c , φ c , φ g , φ g , φ g , φ g ] == [ | G ] (cid:3) g ≤ c ⊙ c (cid:3) c ⊙ c ≤ g (cid:3) g ≤ c ⊙ c (cid:3) c ⊙ c ≤ g [ φ c g , φ c g , φ c , φ g , φ g ] == [ | G ] (cid:3) g ≤ c ⊙ c (cid:3) c ⊙ c ≤ g [ φ c g g , φ c g , φ c g , φ g ] == [ | G ][ φ c g g , φ c g g , φ c g g ] = [ φ g g , φ g g , φ g g ] . It is clear from theorem 33 that any renaming of constants produces a subalgebra: Theorem 34. Let M by an atomized semilattice over C = { c , c , .., c m } . Suppose we aregiven m sets G , G , ...G m with G k = { g k , g k , .., g ke ( k ) } ⊆ G such that G ∩ C = ∅ and the set G = ∪ k G k has cardinal e = | G | . The rename: [ g g ...g e (1) c g g ...g e (2) c ... g m g m ...g me ( m ) c m ] M is isomorphicto the subalgebra of M generated by terms t , t , ..., t e that satisfy: c k is a component constantof t i if and only if g i ∈ G k .Proof. It follows that ( g i ∈ G k ) ⇔ ( c k ∈ C ( t i )) and we can use theorem 33 to state that[ g g ...g e (1) c g g ...g e (2) c ... g m g m ...g me ( m ) c m ] M is the subalgebra of M generated by the terms t , t , ..., t e . Theorem 35. Let M be an atomized semilattice over C = { c , c , .., c m } and A = { φ , φ , .., φ z } a set of atoms of cardinal z = | A | that atomizes M . Let Z be a set of z constants. M is thesubalgebra of F Z ( ∅ ) generated by the terms { t , t , .., t m } over Z defined by: z k is a componentconstant of t i if and only if c i ∈ U c ( φ k ) .Proof. F Z ( ∅ ) is atomized by the same number of atoms than M , each atom with a singleconstant z i in its upper segment. Then M is the rewrite of F Z ( ∅ ) that maps the constants of F Z ( ∅ ) to the upper segments of the atoms of M , as follows: M = [Π zk =1 U c ( φ k ) z k ] F Z ( ∅ ). We cannow use theorem 34 to claim M as a subalgebra of F Z ( ∅ ) generated by the terms t , t , ..., t m over Z , terms that satisfy z k is a component constant of t i if and only if c i ∈ U c ( φ k ).34e can define the product of models M and N , as usual, as the model M ⊗ N withelements ( g, h ) where g is an element of M and h and element of N . The order relation andthe idempotent operator act component-wise, i.e. ( e, f ) ≤ ( g, h ) if and only if e ≤ g and f ≤ h .If M is generated by the constants C M = { a , a , ..., a m } and N is generated by the constants C N = { b , b , ..., b n } , then M ⊗ N has constants ( a i , b j ) and is generated by them.Suppose that C M ∩ C N = ∅ . The element ( x, y ) of M ⊗ N can be mapped, one-to-one, tothe idempotent summation x ⊙ y where x is an element of M and y an element of N , whichworks because e ⊙ f ≤ g ⊙ h if and only if e ≤ g and f ≤ h . We saw before that an element ofthe join model M ⊕ N looks either like an element x of M or like an element y of N or like asummation x ⊙ y . Therefore M ⊗ N is isomorphic to a subalgebra of M ⊕ N , particularly thesubalgebra generated by the terms t ij = a i ⊙ b j . Theorem 36. Let M and N be two semilattice models over the constants C M = { a , a , ..., a m } and C N = { b , b , ..., b n } , respectively, such C M ∩ C N = ∅ . Let G by the set of m × n constants g ij = ( a i , b j ) . The model M ⊗ N generated by G satisfies:i) M ⊗ N = [ | G ]Π mi =1 Π nj =1 (cid:3) g ij ≤ a i ⊙ b j (cid:3) a i ⊙ b j ≤ g ij ( M + N + F G ( ∅ )) .ii) M ⊗ N = [ g , g , ...g ,n a g , g , ...g ,n a ... g m, g m, ...g m,n a m ][ g , g , ...g m, b g , g , ...g m, b ... g ,n g ,n ...g m,n b n ]( M + N ) .iii) M ⊗ N can be atomized with as few non-redundant atoms as the number of non-redundantatoms of M plus the number of non-redundant atoms of N .Proof. (i) Since M ⊗ N is isomorphic to the subalgebra of M ⊕ N generated by the terms t ij = a i ⊙ b j , we can use theorem 33: M ⊗ N = [ | G ]Π mi =1 Π nj =1 (cid:3) g ij ≤ a i ⊙ b j (cid:3) a i ⊙ b j ≤ g ij (( M ⊕ N ) + F G ( ∅ )).Since C M ∩ C N = ∅ it is possible to substitute M ⊕ N by M + N .(ii) This is the result of applying theorem 33(ii) to M ⊕ N = M + N .(iii) Follows directly from proposition (ii), which says that each atom of M ⊗ N is a rename ofan atom in the set M + N .Theorem 36(ii) gives a very simple way to find an atomization for M ⊗ N from atomizationsof M and N .There is a theorem by Birkhoff that says that every algebraic structure A is isomorphic to asubdirect product of subdirectly irreducible algebraic structures that are homomorphic imagesof A (see for example [3, 5]). Stone’s representation theorem can be seen as a particular casefor Boolean algebras [1]. We give now a constructive proof for the celebrated result that anynon-trivial semilattice is a subdirect product of two-element semilattices. In the next theoremwe explicitly build the subdirectly irreducible components and show that each component mapsto an atom of M and is a semilattice with two elements and spawned by two atoms.35 heorem 37. Let M be an atomized semilattice over C = { c , c , .., c m } and A = { φ , φ , .., φ z } a non-empty set of atoms of cardinal z = | A | such that all φ i = ⊖ c and M = [ A ∪ ⊖ c ] . Assume M is not trivial, i.e. M = [ ⊖ c ] . Let B , B , ..., B z a set of z models, such that B j = [ ψ z j , ψ z j z j ] is a model over two constants z j , z j with U c ( ψ z j ) = { z j } and U c ( ψ z j z j ) = { z j , z j } that satisfies B j | = ( z j (cid:12) z j ) and is equal to the homomorphic image of M under the homomorphism: h j : t z j if φ j < t and h j : t z j if φ j < t .i) M is isomorphic to the subalgebra of ⊗ zj =1 B j generated by the constants u c , u c , ..., u c m , eachequal to a tuple u c i = ( u i , u i , ..., u iz ) with u ij = z j if φ j < c i or u ij = z j if φ j < c i .ii) M is isomorphic to the subalgebra of ⊕ zj =1 B j generated by the terms { t , t , .., t m } over Z = ∪ zj =1 { z j , z j } defined by: z k is a component constant of t i if and only if φ k < c i and z k isa component constant of t i if and only if φ k < c i .Proof. (i) Consider an embedding h that maps each constant c i of M to a tuple u c i with z components, that has component j equal to z j when φ j < c i , and equal to z j when φ j < c i .Note that ⊗ zj =1 B j has 2 z constants and that, since ⊙ operates component-wise in a productand z j (cid:12) z j , every summation of constants of ⊗ zj =1 B j produce a constant of ⊗ zj =1 B j . Ourembedding h maps each regular element x of M to a z-tuple u x that is a constant of ⊗ zj =1 B j and has, at position j : either z j if φ j < x or z j if φ j < x , and then ⊗ zj =1 B j | = ( u x ≤ u y ) if andonly if M | = ( x ≤ y ). Therefore, M is isomorphic to the subalgebra of ⊗ zj =1 B j spawned by theconstants u c , u c , ..., u c m , as we wanted to proof.Let us check that it is possible to use theorem 36 to build M as a subalgebra of ⊗ zj =1 B j , i.e. asa rename of B + B + .... + B z . The union B + B + .... + B z is atomized by ∪ zj =1 { ψ z j , ψ z j z j } ,i.e. it is a model with 2 z non-redundant atoms. Let H be the set of 2 z constants of ⊗ zj =1 B j .Each h ∈ H is a z-tuple. The product is then equal to ⊗ zj =1 B j = [Π zj =1 h ∈ H : h j = z j z j h ∈ H : h j = z j z j ][ ∪ zj =1 { ψ z j , ψ z j z j } ] , and the subalgebra of ⊗ zj =1 B j generated by elements u c , u c , ..., u c m named as c , c , .., c m canthen be obtained as: M = [Π z k =1 c i ∈ C : h k = u ci h k ] ⊗ zj =1 B j , a rename that annihilates most constants of ⊗ zj =1 B j . In fact, among the 2 z constants, therename leaves just the m constants u c , u c , ..., u c m . Putting both expressions together: M = [Π zj =1 c i ∈ C : u ij = z j z j c i ∈ C : u ij = z j z j ][ ∪ zj =1 { ψ z j , ψ z j z j } ] , which renames ψ z j to φ j and renames every ψ z j z j to ⊖ c .Note that, without loss of generality (except for excluding the trivial semilattice), we had torequire the atoms of A to satisfy φ i = ⊖ c because, if not, the homomorphism from M to somecomponent B i may not be surjective.(ii) This proposition follows from proposition (i) which says that M is a subalgebra of ⊗ zj =1 B j ,36nd from the fact that ⊗ zj =1 B j is a subalgebra of ⊕ zj =1 B j . To be explicit, we are going to givea couple of constructions of M as a subalgebra of ⊕ zj =1 B j .Consider a construction very similar to the one made for proposition (i) but with z-tuplesreplaced by terms of z components. Each constant c i of M maps to a term t i over 2 Z withcomponent constant z j if φ j < c i , or with component constant z j if φ j < c i . Since ⊙ isassociative and commutative and taking into account that z j (cid:12) z j then ⊙ operates over theterms t i producing elements that can be represented with a term with z component constants.For each regular element x of M , the idempotent summation, ⊙ i : c i ≤ x t i that runs along every i such that c i ≤ x produces a term t x over 2 Z with exactly z component constants, a termthat has either component constant z j if φ j < x or component constant z j if φ j < x . It followsthat ⊕ zj =1 B j | = ( t x ≤ t y ) if and only if M | = ( x ≤ y ), so M is isomorphic to the subalgebragenerated by the terms { t , t , .., t m } which proves the theorem.Here is another explicit construction: Since the constants of the different B j are all pairwisedisjoint ⊕ zj =1 B j = B + B + .... + B z = [ ∪ zj =1 { ψ z j , ψ z j z j } ] and M is the rename: M = [Π zj =1 c i ∈ C : φ j Humane AI; Toward AI Systems That Augmentand Empower Humans by Understanding Us, our Society and the World Around Us (grant ALMA: Human Centric Algebraic Machine Learning (grant Notation and basic definitions The reader should be familiar with the following notation and definitions:We use a minus symbol for the subtraction of sets, i.e. we use A − B instead of A \ B .We use capital letters such as M or N to refer to semilattice models or to refer to atomizedsemilattice models. We use small caps to refer to elements of a model. We sometimes use thesame letter for a model and the set of atoms of the model. We use Greek letters to refer toatoms.A term is a recipe to form an element by using the constants and the idempotent operator,for example t = c ⊙ c ⊙ c . Multiple terms can yield the same element in a semilattice model.Two terms s and t yield the same element in a model M if M | = ( s ≤ t ) ∧ ( t ≤ s ).Atomized semilattices have elements of two sorts, the regular elements and the atoms.Regular elements form a semilattice. Regular elements and atoms form a partial order.The free model (or freest model) F C ( ∅ ) over C is the model with a different element foreach term over C . We sometimes refer to elements of F C ( ∅ ) with the word “term”. Terms areeither constants or idempotent summations of constants formed using the idempotent, binaryoperator ⊙ . When a term is equal to a single constant we usually refer to it with the word‘constant”. Two terms are equal in the freest model if and only if they can be proven equal byusing the commutative, associative and idempotent properties, so c ⊙ c ⊙ c is the same termthan c ⊙ c .Each regular element of an atomized semilattice model M over a set of constants C cor-responds to an equivalence class in F C ( ∅ ). Each atom element of M is defined by a subset ofconstants taken from C .We use bold to represent functions that yield sets. We use C ( M ) for the constants and and A ( M ) for the atoms of a model M . The lower and upper segments of an element x are defined,asymmetrically, as L M ( x ) = { y : y < x ∨ y = x } and U M ( x ) = { y : y > x } . The superscript a is used to denote the intersection with the atoms: L aM ( x ) = L M ( x ) ∩ A ( M ). We also usethe superscript c to represent the intersection with the constants L cM ( x ) = L M ( x ) ∩ C ( M ) and U cM ( x ) = U M ( x ) ∩ C ( M ).In an atomized semilattice model, U cM ( φ ) is defined as the set of constants in the uppersegment of the atom φ . Since an atom is defined by the constants in its upper segment weoften drop the subindex M and simply write U c ( φ ). Atoms can be identified across models byits upper constant segment.We say an atom φ is wider than atom η if it is different than η and for every constant c ,39 η < c ) ⇒ ( φ < c ). Equivalently an atom φ is wider than atom η when U c ( η ) ( U c ( φ ).An atom φ is redundant with model M iff for each constant c such that φ < c there is atleast one atom η < c in M such that φ is wider than η . An atom of M that is not redundantwith M is called a “non-redundant” atom of M .We say an atom φ is “in M ” or is “an atom of M ” and write φ ∈ M if, there is a set ofatoms A such that M = [ A ∪ { φ } ].An atom is “external” to M , written φ M , if it is not in M . With respect to a model M an atom can be either a ‘non-redundant atom in M or redundant with M or external to M .The component constants C ( a ) of a constant or term is a set defined as C ( c ) = { c } if c isa constant and if t is a term t = c ⊙ c ⊙ ...c n − ⊙ c n , as the set C ( t ) = { c , c , ..., c n } . Atomshave no component constants.We use the word ”duple” to refer to an ordered pair of elements r ≡ ( a, b ) of F C ( ∅ ). Wesay a model M satisfies the positive duple r + if M | = ( a ≤ b ) and satisfies the negative duple r − if M = ( a ≤ b ) or, equivalently, M | = ( a b ).A set of positive duples R + of a model M is a set of duples that are satisfied by M. Whenwe write M | = R + we mean M | = r + for all r ∈ R + . In the same way, when we write M | = R − we mean M | = r − for all r ∈ R − .The theory of a model M , written T h ( M ), is the set of all first order sentences that aretrue in M , and T h ( M ) is used to refer to atomic sentences without quantifiers that are truein M . We use here T h ( M ) as the set of positive and negative duples satisfied by the M . Thepositive and negative theories of M , written T h +0 ( M ) and T h − ( M ) are, respectively, the sets ofduples satisfied by M (positive duples of M ) and the set of duples not satisfied by M (negativeduples of M ).We say element y contains x if x ≤ y .The atom ⊖ c is the atom defined by U c ( ⊖ c ) = C .When A is a set of atoms we often refer to the model spawned by the atoms in A simpleas A , and sometimes by using brackets [ A ].The discriminant dis M ( a, b ) in an atomized model M is the set L aM ( a ) − L aM ( b ) of atomsof M . The duple a ≤ b holds in M if and only if dis M ( a, b ) is empty.For each atom φ = ⊖ c , the “pinning term” T φ is equal to the idempotent summation of allthe constants in the set C − U c ( φ ). For each constant c ∈ U c ( φ ) define the “pinning duple”as the negative atomic sentence ( c < T φ ). P R ( φ ) is defined as the set of pinning duples of anatom φ . 40e often used R to refer to a set of positive and/or negative duples, i.e. a set of dupleseach with its own sign (signed duples). We use R + and R − for the positive and the negativeduples in the set so R = R + ∪ R − .A model A over the constants C is “freer or as free” then model B over the constants K ⊆ C if for every duple r over K , B | = r − ⇒ A | = r − . In other words, T h − ( B ) ⊆ T h − ( A ).We sometimes use the word freer for short to mean freer or as free.The freest model F C ( R + ) is the model such that if F C ( R + ) | = r + for some duple r thenevery model of R + also satisfies r + . When every model of R + satisfies r + we write R + ⇒ r + or R + ⊢ r + .We use A + B or [ A ∪ B ] to refer to the model spawned by the atoms of A and the atomsof B . ▽ is the “union” operator of atoms. The union of atoms is an operation that produces anatom φ ▽ ψ such that U c ( φ ▽ ψ ) ≡ U c ( φ ) ∪ U c ( ψ ).The “full crossing”, (cid:3) r M , of a duple r = ( r L , r R ) in an atomized model M such that M | = r − is a mechanism that creates another model N | = T h +0 ( M ) ∪ { r + } . To calculate thefull crossing, define the sets H = dis M ( r ) and B = L aM ( r R ) and obtain the model atomizedby the set N = ( M − H ) ∪ ( H ▽ B ) where the letter M has been used to represent the set ofatoms of model M and where H ▽ B ≡ { λ ▽ ρ : ( λ ∈ H ) ∧ ( ρ ∈ B ) } .For models M and N we write M ⊆ N if for every atom φ ∈ M also holds φ ∈ N . References [1] Marshall Harvey Stone. The theory of representation for boolean algebras. Trans. Amer.Math. Soc. , 40:37–11, 1936.[2] Fernando Martin-Maroto and Gonzalo G. de Polavieja. Algebraic machine learning. arXiv:1803.05252 , 2018.[3] Stanley Burris and H. P. Sankappanavar. A course in universal algebra . Springer-Verlag,1981.[4] Dona Papert. Congruence relations in semi-lattices. Journal of the London MathematicalSociety , s1-39(1):723–729, Jan, 1964.[5] Garrett Bikhoff. Subdirect products in universal algebra.