aa r X i v : . [ m a t h . L O ] J a n Absolutely Free Hyperalgebras
Coniglio, Marcelo E. ∗ and Toledo, Guilherme V. † Institute of Philosophy and the Humanities - IFCH andCentre for Logic, Epistemology and The History of Science - CLEUniversity of Campinas - UnicampCampinas, SP, Brazil
January 12, 2021
Abstract
It is well know from universal algebra that, for every signature Σ , there exist algebras over Σ which are freelygenerated. Furthermore, they are, up to isomorphisms, unique, and equal to algebras of terms. Equivalently, theforgetful functor, from the category of Σ -algebras to Set , has a left adjoint.This result does not extend to hyperalgebras, which generalize algebras by allowing the result of an operationto assume a non-empty set of values. Not only freely generated hyperalgebras do not exist, but the forgetfulfunctor , from the category of Σ -hyperalgebras to Set , does not have a left adjoint.In this paper we generalize, in a natural way, algebras of terms to hyperalgebras of terms, which display manyproperties of freely generated algebras: they extend uniquely to homomorphisms, not functions, but pairs offunctions and collections of choices, which select how an homomorphism approaches indeterminacies; and theyare generated by a set that fits a strong definition of basis, which we call the ground of the hyperalgebra. Withthese definitions at hand, we offer simplified proofs that freely generated hyperalgebras do not exist and that does not have a left adjoint. Keywords:
Freely generated algebras, absolutely free algebras, hyperalgebras, multialgebras, non-deterministicalgebras, category of hyperalgebras.
Contents (Σ , , 𝜅 ) as. . . 4 cdf -generated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 . . . being disconnected and generated by its ground . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 . . . being disconnected and having a strong basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 . . . being disconnected and chainless . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 ∗ [email protected] † [email protected] In the realm of universal algebra, it is a well know result ([1]) that there exist algebras over a given signature Σ that are freely generated by subsets 𝑋 of their universe 𝐴 , meaning that for any other Σ -algebra , with universe 𝐵 ,and function 𝑓 ∶ 𝑋 → 𝐵 , there exists precisely one homomorphism 𝑓 between and that extends 𝑓 ; moreover,the Σ -algebra freely generated by 𝑋 is isomorphic to the Σ -algebra of terms over 𝑋 , that is T (Σ , 𝑋 ) , and thereforeunique up to isomorphisms. Equivalently, in the language of categories, the forgetful functor 𝑈 ∶ Alg (Σ) → Set ,from the category of Σ -algebras to the category of sets, has a left adjoint 𝐹 , associating to a set 𝑋 any Σ -algebrafreely generated by 𝑋 .These objects do not extend well to the context of hyperalgebras: this latter notion, also known as multialgebrasor non-deterministic algebras, introduced in [2], generalizes the concept of an algebra by replacing operations bymultioperations, whose results assume multiple values, that is, a subset of the universe; here, we will restrictourselves to hyperalgebras whose operations cannot return an empty set of values, which is a common requirementwhen working with non-classical logics and their semantics. It is easy to prove that, first of all, freely generatedhyperalgebras do not exist, and second, that the forgetful functor ∶ MAlg (Σ) → Set , from the category ofhyperalgebras over the signature Σ to the category of sets, does not have a left adjoint.We have reasons to believe that understanding as terms, in the category of hyperalgebras, only those elementsfound in the algebras of terms as classically defined (namely, in the sets T (Σ , ) ) disregards other hyperalgebraswith an astoundingly similar behaviour, so that we generalize the algebras of terms to hyperalgebras of terms. Suchobjects indeed present many of the properties expected of the algebra of terms and, equivalently, freely generatedalgebras:1. they extend uniquely to homomorphisms, not functions, but pairs of functions and what we will call collec-tions of choices, that “select” how an homomorphism will approach indeterminacies;2. they are somewhat “free” of identities, a generic intuition we formalize by defining disconnected hyperal-gebras, and they can be generated by a set of “indecomposable” elements we shall call the ground of thehyperalgebra, much like variables, which are terms without proper subterms;3. strengthening the previous point, they are not only disconnected but have a minimum generating set thatbehaves quite similarly to a basis, of exempli gratia a vector space, which we can prove to be precisely theground;4. strengthening even further the existence of the ground, they are disconnected and every sequence of fur-ther simpler and simpler elements eventually ends on an indecomposable element, condition we call being“chainless”.What is more impressive, we prove all these characterizations are equivalent; and with these objects at hand,we will offer simplified proofs that freely generated hyperalgebras do not exist and that does not have a leftadjoint. We will understand as a signature a collection
Σ = {Σ 𝑛 } 𝑛 ∈ ℕ of sets Σ 𝑛 not necessarily non-empty, and will denoteby Σ either the collection itself or, when there is no risk of confusion, the union ⋃ 𝑛 ∈ ℕ Σ 𝑛 .A Σ -hyperalgebra, or multialgebra, is a pair = ( 𝐴, { 𝜎 } 𝜎 ∈Σ ) , where 𝐴 is a non-empty set and { 𝜎 } 𝜎 ∈Σ is acollection of functions indexed by Σ (the union), such that, if 𝜎 ∈ Σ 𝑛 , 𝜎 is a function of the form 𝜎 ∶ 𝐴 𝑛 → ( 𝐴 ) ⧵ {∅} , that is, an 𝑛 − ary function from 𝐴 to the set of non-empty parts of 𝐴 .An homomorphism between two Σ -hyperalgebras = ( 𝐴, { 𝜎 } 𝜎 ∈Σ ) and = ( 𝐵, { 𝜎 } 𝜎 ∈Σ ) is a function 𝑓 ∶ 𝐴 → 𝐵 such that, for all 𝑛 ∈ ℕ , 𝜎 ∈ Σ 𝑛 and elements 𝑎 , … , 𝑎 𝑛 ∈ 𝐴 , { 𝑓 ( 𝑎 ) ∶ 𝑎 ∈ 𝜎 ( 𝑎 , … , 𝑎 𝑛 )} ⊆ 𝜎 ( 𝑓 ( 𝑎 ) , … , 𝑓 ( 𝑎 𝑛 )); when we have, in the previous equation, instead of an inclusion an equality, we say 𝑓 is a full homomorphism. If thehomomorphism 𝑓 ∶ 𝐴 → 𝐵 is injective, we call it a monomorphism ; if it is surjective, we call it an epimorphism ;and if it is a bijective full homomorphism, we call it an isomorphism . To signify that 𝑓 is an homomorphism from to , we write 𝑓 ∶ → .bsolutely Free Hyperalgebras 3The class of all Σ -hyperalgebras, equipped with the homomorphisms between them, becomes the category MAlg (Σ) . In this category, the epics are precisely the epimorphisms, while any monomorphism is a monic. Inturn, isomorphims, as defined above, are exactly the isomorphisms in the categorical sense (see, for instance, [4],section 2). Notice, however, that is not known whether all monics are monomorphisms.Given two Σ -hyperalgebras = ( 𝐴, { 𝜎 } 𝜎 ∈Σ ) and = ( 𝐵, { 𝜎 } 𝜎 ∈Σ ) such that 𝐵 ⊆ 𝐴 , we say is asubhyperalgebra of if the identity function 𝑖𝑑 ∶ 𝐵 → 𝐴 is an homomorphism from to , that is, 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) ⊆ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) . Given a set of variables and a signature Σ = {Σ 𝑛 } 𝑛 ∈ ℕ , the algebra of terms freely generated by over Σ will bedenoted T (Σ , ) , and its universe will be denoted 𝑇 (Σ , ) . Intuitively, the set of terms 𝑇 (Σ , ) is the smallest setcontaining:1. the variables ;2. the expression 𝜎𝛼 … 𝛼 𝑛 , given a 𝜎 ∈ Σ 𝑛 and already defined terms 𝛼 , … , 𝛼 𝑛 in T (Σ , ) .The set 𝑇 (Σ , ) becomes the Σ -algebra T (Σ , ) when we define, for a 𝜎 ∈ Σ 𝑛 and terms 𝛼 , … , 𝛼 𝑛 in T (Σ , ) , 𝜎 T (Σ , ) ( 𝛼 , … , 𝛼 𝑛 ) = 𝜎𝛼 … 𝛼 𝑛 . We define the order (or complexity, or depth) 𝗈 ( 𝛼 ) of a term 𝛼 of T (Σ , ) as: 𝗈 ( 𝛼 ) = if 𝛼 ∈ ∪ Σ ; and 𝗈 ( 𝜎𝛼 … 𝛼 𝗇 ) = + max{ 𝗆 , … , 𝗆 𝗇 } , if 𝗈 ( 𝛼 𝗃 ) = 𝗆 𝗃 for ≤ 𝑗 ≤ 𝑛 . Definition 2.1.
Given a signature Σ and a cardinal 𝜅 > , the expanded signature Σ 𝜅 = {Σ 𝜅𝑛 } 𝑛 ∈ ℕ is the signaturesuch that Σ 𝜅𝑛 = Σ 𝑛 × 𝜅 , where we will denote the pair ( 𝜎, 𝛽 ) by 𝜎 𝛽 for 𝜎 ∈ Σ and 𝛽 ∈ 𝜅 . We demand that 𝜅 is greater than zero: hence, if Σ is non-empty, so is Σ 𝜅 . Definition 2.2.
Given a set of variables , a signature Σ and a cardinal 𝜅 > , we define the Σ -hyperalgebra ofnon-deterministic terms, or simply hyperalgebra of terms, as mT (Σ , , 𝜅 ) = ( 𝑇 (Σ 𝜅 , ) , { 𝜎 mT (Σ , ,𝜅 ) } 𝜎 ∈Σ ) with universe 𝑇 (Σ 𝜅 , ) and such that, for 𝜎 ∈ Σ 𝑛 and 𝛼 , … , 𝛼 𝑛 ∈ 𝑇 (Σ 𝜅 , ) , 𝜎 mT (Σ , ,𝜅 ) ( 𝛼 , … , 𝛼 𝑛 ) = { 𝜎 𝛽 𝛼 … 𝛼 𝑛 ∶ 𝛽 ∈ 𝜅 } . The intuition behind this definition is that, connecting given terms 𝛼 through 𝛼 𝑛 with a connective 𝜎 can, in abroader interpretation taking into account non-determinism, return many terms with the same general shape, thatis 𝜎𝛼 … 𝛼 𝑛 , over which we maintain certain degree of control by counting them, what we achieve by using anindex to our connective, 𝜎 𝛽 .One can ask why all connectives must return the exact same number of generalized terms, that is 𝜅 , but this willnot be the case: more useful to us shall be the subhyperalgebras of mT (Σ , , 𝜅 ) , where the cardinality will vary aslong as it is bounded by 𝜅 ; we have defined the hyperalgebras of terms as above since defining its subhyperalgebrasdirectly is substantially more difficult.Here, we will restrict ourselves to the cases where Σ ≠ ∅ or ≠ ∅ , so that mT (Σ , , 𝜅 ) is always well defined.We will understand as the order of an element 𝛼 of mT (Σ , , 𝜅 ) simply its order as an element of 𝑇 (Σ 𝜅 , ) .Notice that, if 𝜎 mT (Σ , ,𝜅 ) ( 𝛼 , … , 𝛼 𝑛 ) ∩ 𝜃 mT (Σ , ,𝜅 ) ( 𝛽 , … , 𝛽 𝑚 ) ≠ ∅ , then 𝜎 = 𝜃 , 𝑛 = 𝑚 and 𝛼 = 𝛽 , . . . , 𝛼 𝑛 = 𝛽 𝑚 , since if the intersection is not empty there are 𝛽, 𝛾 ∈ 𝜅 such that 𝜎 𝛽 𝛼 … 𝛼 𝑛 = 𝜃 𝛾 𝛽 … 𝛽 𝑚 and by the structure of 𝑇 (Σ 𝜅 , ) we find that 𝜎 𝛽 = 𝜃 𝛾 . Example 2.3.
The Σ -algebras of terms T (Σ , ) , when considered as hyperalgebras such that 𝜎 T (Σ , ) ( 𝛼 , … , 𝛼 𝑛 ) ={ 𝜎𝛼 … 𝛼 𝑛 } , are hyperalgebras of terms, with 𝜅 = 1 ; that is, T (Σ , ) and mT (Σ , , are isomorphic. From now on, the cardinal of a set 𝑋 will be denoted by | 𝑋 | .bsolutely Free Hyperalgebras 4 Example 2.4.
A directed graph is a pair ( 𝑉 , 𝐴 ) , with 𝑉 a non-empty set of elements called vertices and 𝐴 ⊆ 𝑉 aset of elements called arrows, where we say that there is an arrow from 𝑢 to 𝑣 , both in 𝑉 , if ( 𝑢, 𝑣 ) ∈ 𝐴 ; we say thatthe 𝑛 -tuple ( 𝑛 , … , 𝑣 𝑛 ) is a path between 𝑢 and 𝑣 if 𝑢 = 𝑣 , 𝑣 = 𝑣 𝑛 and ( 𝑣 𝑖 , 𝑣 𝑖 +1 ) ∈ 𝐴 for every 𝑖 ∈ {1 , … , 𝑛 − 1} ;we say that 𝑢 ∈ 𝑉 has a successor if there exists 𝑣 ∈ 𝑉 such that ( 𝑢, 𝑣 ) ∈ 𝐴 , and 𝑢 has a predecessor if there exists 𝑣 ∈ 𝑉 such that ( 𝑣, 𝑢 ) ∈ 𝐴 .A directed graph 𝐹 = ( 𝑉 , 𝐴 ) is a forest if, for any two 𝑢, 𝑣 ∈ 𝑉 , there exists at most one path between 𝑢 and 𝑣 ,and a forest is said to have height 𝜔 if every vertex has a successor. Then, we state that forests of height 𝜔 are inbijection with the subhyperalgebras of the hyperalgebras of terms over the signature Σ 𝑠 with exactly one operator 𝑠 of arity .Essentially, take as the set of elements of 𝐹 that have no predecessor and define, for 𝑢 ∈ 𝑉 , 𝑠 ( 𝑢 ) = { 𝑣 ∈ 𝑉 ∶ ( 𝑢, 𝑣 ) ∈ 𝐴 } , and we have that the Σ 𝑠 -hyperalgebra = ( 𝑉 , { 𝑠 }) , subhyperalgebra of mT (Σ 𝑠 , , | 𝑉 | ) , carries the same infor-mation that 𝐹 . Example 2.5.
More generally, a directed multi-graph [5], or directed 𝑚 -graph, is a pair ( 𝑉 , 𝐴 ) with 𝑉 a non-empty set of vertices and 𝐴 a subset of 𝑉 + × 𝑉 , where 𝑉 + = ⋃ 𝑛 ∈ ℕ ⧵ {0} 𝑉 𝑛 is the set of finite, non-empty, sequencesover 𝑉 . We will say that ( 𝑣 , … , 𝑣 𝑛 ) is a path between 𝑢 and 𝑣 if 𝑢 = 𝑣 , 𝑣 = 𝑣 𝑛 and, for every 𝑖 ∈ {1 , … , 𝑛 − 1} ,there exists 𝑣 𝑖 , … , 𝑣 𝑖 𝑚 such that (( 𝑣 𝑖 , … , 𝑣 𝑖 𝑚 ) , 𝑣 𝑖 +1 ) , with 𝑣 𝑖 = 𝑣 𝑖 𝑗 for some 𝑗 ∈ {1 , … , 𝑚 } .Then an 𝑚 -forest is a directed 𝑚 -graph such that any two elements are connected by at most one path; and an 𝑚 -forest is said to have 𝑛 -height 𝜔 , for 𝑛 ∈ ℕ ⧵ {0} , if, for any ( 𝑢 , … , 𝑢 𝑛 ) ∈ 𝑉 𝑛 , there exists 𝑣 ∈ 𝑉 such that (( 𝑢 , … , 𝑢 𝑛 ) , 𝑣 ) ∈ 𝐴 . Finally, we see that every 𝑚 -forest 𝐹 = ( 𝑉 , 𝐴 ) with 𝑛 -height 𝜔 , for every 𝑛 ∈ 𝑆 ⊆ ℕ ⧵ {∅} ,is essentially equivalent to the Σ 𝑆 -hyperalgebra = ( 𝑉 , { 𝜎 } 𝜎 ∈Σ 𝑆 ) , with 𝜎 ( 𝑢 , … , 𝑢 𝑛 ) = { 𝑣 ∈ 𝑉 ∶ (( 𝑢 , … , 𝑢 𝑛 ) , 𝑣 ) ∈ 𝐴 } , for 𝜎 of arity 𝑛 , and Σ 𝑆 the signature with exactly one operator of arity 𝑛 , for every 𝑛 ∈ 𝑆 . It is not hard to see that is a subhyperalgebra of mT (Σ 𝑆 , , | 𝑉 | ) , with the set of elements 𝑣 of 𝑉 such that, for no ( 𝑢 , … , 𝑢 𝑛 ) ∈ 𝑉 + , (( 𝑢 , … , 𝑢 𝑛 ) , 𝑣 ) ∈ 𝐴 . (Σ , , 𝜅 ) as. . . Now, in universal algebra, the algebras of terms T (Σ , ) are absolutely free, also said to be freely generated, alsosaid to be freely generated in the variety of all Σ -algebras: this means that there exists a set, in their case the set ofvariables , such that, for every other Σ -algebra with universe 𝐵 and function 𝑓 ∶ → 𝐵 , there exists an uniquehomomorphism 𝑓 ∶ T (Σ , ) → extending 𝑓 . As we mentioned before, this is no longer true when dealing withhyperalgebras, but we can define a closely related concept with the aid of what we will call collections of choices .Collections of choices are motivated by the notion of legal valuations , first defined in Avron and Lev’s seminalpaper [3] in the context of non-deterministic logical matrices. A map 𝜈 from T (Σ , ) (seen as the algebra ofpropositional formulas over Σ freely generated by ) to the universe of a Σ -hyperalgebra is a legal valuationwhenever 𝜈 ( 𝜎𝛼 … 𝛼 𝑛 ) ∈ 𝜎 ( 𝜈 ( 𝛼 ) , … , 𝜈 ( 𝛼 𝑛 )) , for any connective 𝜎 in Σ ; essentially, at a formula 𝜎𝛼 … 𝛼 𝑛 , 𝜈 “chooses” a value from all the possible values 𝜎 ( 𝜈 ( 𝛼 ) , … , 𝜈 ( 𝛼 𝑛 )) , possible values which depend themselves onthe previous choices 𝜈 ( 𝛼 ) through 𝜈 ( 𝛼 𝑛 ) performed by 𝜈 .What a collection of choices does is then to automatize this choices, which justifies its name. Definition 3.1.
Given hyperalgebras = ( 𝐴, { 𝜎 } 𝜎 ∈Σ ) and = ( 𝐵, { 𝜎 } 𝜎 ∈ 𝜎 ) over the signature Σ , a collectionof choices from to is a collection 𝐶 = { 𝐶 𝑛 } 𝑛 ∈ ℕ of collections of functions 𝐶 𝑛 = { 𝐶𝜎 𝑏 ,...,𝑏 𝑛 𝑎 ,...,𝑎 𝑛 ∶ 𝜎 ∈ Σ 𝑛 , 𝑎 , ..., 𝑎 𝑛 ∈ 𝐴, 𝑏 , ..., 𝑏 𝑛 ∈ 𝐵 } such that, for 𝜎 ∈ Σ 𝑛 , 𝑎 , ..., 𝑎 𝑛 ∈ 𝐴 and 𝑏 , ..., 𝑏 𝑛 ∈ 𝐵 , 𝐶𝜎 𝑏 ,...,𝑏 𝑛 𝑎 ,...,𝑎 𝑛 is a function of the form 𝐶𝜎 𝑏 ,...,𝑏 𝑛 𝑎 ,...,𝑎 𝑛 ∶ 𝜎 ( 𝑎 , ..., 𝑎 𝑛 ) → 𝜎 ( 𝑏 , ..., 𝑏 𝑛 ) . Example 3.2. If is actually an algebra, that is, all its operations return singletons, there only exists one collectionof choices from any to ; this means that in the classical environment of universal algebras, collections of choicesare somewhat irrelevant. bsolutely Free Hyperalgebras 5 Example 3.3.
A directed tree is a directed forest where there exists exactly one element without predecessor; wesay that 𝑣 ramifies from 𝑢 if there exists an arrow from 𝑢 to 𝑣 . Then, given two directed trees 𝑇 = ( 𝑉 , 𝐴 ) and 𝑇 = ( 𝑉 , 𝐴 ) of height 𝜔 , seem as Σ 𝑠 -hyperalgebras, and a collection of choices 𝐶 from 𝑇 to 𝑇 , for every 𝑣 ∈ 𝑉 and 𝑢 ∈ 𝑉 the function 𝐶𝑠 𝑢𝑣 chooses, for each of the elements that ramify from 𝑣 , one element that ramifies from 𝑢 . Definition 3.4.
Given a signature Σ , a Σ -hyperalgebra = ( 𝐴, { 𝜎 } 𝜎 ∈Σ ) is choice-dependent freely generatedby 𝑋 if 𝑋 ⊆ 𝐴 and, for all Σ -hyperalgebras = ( 𝐵, { 𝜎 } 𝜎 ∈Σ ) , all functions 𝑓 ∶ 𝑋 → 𝐵 and all collections ofchoices 𝐶 from to , there is a unique homomorphism 𝑓 𝐶 ∶ → such that:1. 𝑓 𝐶 | 𝑋 = 𝑓 ;2. for all 𝜎 ∈ Σ 𝑛 and 𝑎 , … , 𝑎 𝑛 ∈ 𝐴 , 𝑓 𝐶 | 𝜎 ( 𝑎 , … ,𝑎 𝑛 ) = 𝐶𝜎 𝑓 𝐶 ( 𝑎 ) , … ,𝑓 𝐶 ( 𝑎 𝑛 ) 𝑎 , … ,𝑎 𝑛 . For simplicity, when is choice-dependent freely generated by 𝑋 , we will write that is cdf -generated by 𝑋 . In the next definition, we introduce the concept of ground to indicate what elements of a hyperalgebra are not“achieved” by its multioperations; alternatively, while thinking of terms and their respective algebras, the groundis the set of indecomposable terms, that is, variables. Definition 3.5.
Given a Σ -hyperalgebra = ( 𝐴, { 𝜎 } 𝜎 ∈Σ ) , we define its build as 𝐵 ( ) = ⋃ { 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) ∶ 𝑛 ∈ ℕ , 𝜎 ∈ Σ 𝑛 , 𝑎 , … , 𝑎 𝑛 ∈ 𝐴 } . We define the ground of as 𝐺 ( ) = 𝐴 ⧵ 𝐵 ( ) . Example 3.6. 𝐵 ( T (Σ , )) = 𝑇 (Σ , ) ⧵ and 𝐺 ( T (Σ , )) = . Example 3.7. If 𝐹 = ( 𝑉 , 𝐴 ) is a directed tree of height 𝜔 , thought as a Σ 𝑠 -hyperalgebra, its ground is the set ofelements 𝑣 in 𝑉 without predecessors. Proposition 3.8.
1. If 𝑓 ∶ → is an homomorphism between Σ -hyperalgebras, then 𝐵 ( ) ⊆ 𝑓 −1 ( 𝐵 ( )) and 𝑓 −1 ( 𝐺 ( )) ⊆ 𝐺 ( ) ;2. If is a subhyperalgebra of , 𝐵 ( ) ⊆ 𝐵 ( ) and 𝐺 ( ) ⊆ 𝐺 ( ) .Proof.
1. If 𝑎 ∈ 𝐵 ( ) , there exist 𝜎 ∈ Σ 𝑛 and 𝑎 , … , 𝑎 𝑛 ∈ 𝐴 such that 𝑎 ∈ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) . Since 𝑓 ( 𝜎 ( 𝑎 , … , 𝑎 𝑛 )) ⊆ 𝜎 ( 𝑓 ( 𝑎 ) , … , 𝑓 ( 𝑎 𝑛 )) , we find that 𝑓 ( 𝑎 ) ∈ 𝜎 ( 𝑓 ( 𝑎 ) , … , 𝑓 ( 𝑎 𝑛 )) and therefore 𝑓 ( 𝑎 ) ∈ 𝐵 ( ) , meaning that 𝑎 ∈ 𝑓 −1 ( 𝐵 ( )) . Using that 𝐺 ( ) = 𝐴 ⧵ 𝐵 ( ) we obtain the second mentioned inclusion.2. If 𝑏 ∈ 𝐵 ( ) , there exist 𝜎 ∈ Σ 𝑛 and 𝑏 , … , 𝑏 𝑛 ∈ 𝐵 such that 𝑏 ∈ 𝜎 ( 𝑏 , … , 𝑏 𝑛 ) , and given that 𝜎 ( 𝑏 , … , 𝑏 𝑛 ) ⊆ 𝜎 ( 𝑏 , … , 𝑏 𝑛 ) we obtain 𝑏 ∈ 𝐵 ( ) . Using again that 𝐺 ( ) = 𝐴 ⧵ 𝐵 ( ) we finish theproof.From this it also follows that if 𝑓 ∶ → is an homomorphism, 𝐺 ( ) ∩ 𝑓 ( 𝐴 ) is contained in { 𝑓 ( 𝑎 ) ∶ 𝑎 ∈ 𝐺 ( )} . Indeed, if 𝑏 is in 𝐺 ( ) ∩ 𝑓 ( 𝐴 ) , 𝑎 ∈ 𝐴 such that 𝑓 ( 𝑎 ) = 𝑏 is in 𝑓 −1 ( 𝐺 ( )) and, by the previous proposition,it is in 𝐺 ( ) , and therefore 𝑏 is in { 𝑓 ( 𝑎 ) ∶ 𝑎 ∈ 𝐺 ( )} .Generalizing 3.6, we have that 𝐺 ( mT (Σ , , 𝜅 )) = : we proceed by induction to show that 𝐵 ( mT (Σ , , 𝜅 )) = 𝑇 (Σ 𝜅 , ) ⧵ , what is equivalent. If 𝛼 is of order , either we have 𝛼 = 𝜎 𝛽 , for a 𝜎 ∈ Σ and 𝛽 ∈ 𝜅 , and therefore 𝛼 ∈ 𝐵 ( mT (Σ , , 𝜅 )) ; or we have that 𝛼 = 𝑝 ∈ , and if there exists 𝜎 ∈ Σ 𝑚 and 𝛼 , … , 𝛼 𝑚 ∈ 𝑇 (Σ 𝜅 , ) such that 𝑝 ∈ 𝜎 mT (Σ , ,𝜅 ) ( 𝛼 , … , 𝛼 𝑚 ) we have 𝑝 = 𝜎 𝛽 𝛼 … 𝛼 𝑚 for 𝛽 ∈ 𝜅 , which is absurd given the structure of 𝑇 (Σ 𝜅 , ) , forcing us to conclude that 𝑥 ∉ 𝐵 ( mT (Σ , , 𝜅 )) . If 𝛼 is of order 𝑛 , we have that 𝛼 = 𝜎 𝛽 𝛼 … 𝛼 𝑚 for a 𝜎 ∈ Σ 𝑚 , 𝛽 ∈ 𝜅 and 𝛼 , … , 𝛼 𝑚 of orderat most 𝑛 − 1 , and therefore we have 𝛼 in 𝜎 mT (Σ , ,𝜅 ) ( 𝛼 , … , 𝛼 𝑚 ) , meaning that 𝛼 ∈ 𝐵 ( mT (Σ , , 𝜅 )) .bsolutely Free Hyperalgebras 6 Definition 3.9.
Given a Σ -hyperalgebra = ( 𝐴, { 𝜎 } 𝜎 ∈Σ ) and a set 𝑆 ⊆ 𝐴 , we define the sets ⟨ 𝑆 ⟩ 𝑚 by induction: ⟨ 𝑆 ⟩ = 𝑆 ∪ ⋃ 𝜎 ∈Σ 𝜎 ; and assuming we have defined ⟨ 𝑆 ⟩ 𝑚 , we make ⟨ 𝑆 ⟩ 𝑚 +1 = ⟨ 𝑆 ⟩ 𝑚 ∪ ⋃ { 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) ∶ 𝑛 ∈ ℕ , 𝜎 ∈ Σ 𝑛 , 𝑎 , … , 𝑎 𝑛 ∈ ⟨ 𝑆 ⟩ 𝑚 } . The set generated by 𝑆 , denoted by ⟨ 𝑆 ⟩ , is then defined as ⟨ 𝑆 ⟩ = ⋃ 𝑚 ∈ ℕ ⟨ 𝑆 ⟩ 𝑚 .We say is generated by 𝑆 if ⟨ 𝑆 ⟩ = 𝐴 . Lemma 3.10.
Every subhyperalgebra of mT (Σ , , 𝜅 ) is generated by 𝐺 ( ) .Proof. Suppose 𝑎 is an element of not contained in ⟨ 𝐺 ( ) ⟩ of minimum order: since 𝑎 cannot belong to 𝐺 ( ) ∪ ⋃ 𝜎 ∈Σ 𝜎 = ⟨ 𝐺 ( ) ⟩ , there exist 𝑛 > , 𝜎 ∈ Σ 𝑛 and 𝑎 , … , 𝑎 𝑛 ∈ 𝐴 such that 𝑎 ∈ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) .Since 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) ⊆ 𝜎 mT (Σ ,𝑋,𝜅 ) ( 𝑎 , … , 𝑎 𝑛 ) we derive that 𝑎 to 𝑎 𝑛 are of smaller order than 𝑎 : by our hy-pothesis, there must exist 𝑚 , … , 𝑚 𝑛 such that 𝑎 𝑗 ∈ ⟨ 𝐺 ( ) ⟩ 𝑚 𝑗 for all 𝑗 ∈ {1 , … , 𝑛 } ; taking 𝑚 = max{ 𝑚 , … , 𝑚 𝑛 } , 𝑎 , … , 𝑎 𝑛 ∈ ⟨ 𝐺 ( ) ⟩ 𝑚 , and therefore 𝑎 ∈ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) ⊆ ⟨ 𝐺 ( ) ⟩ 𝑚 +1 , which contradicts our assumption and proves the lemma. Theorem 3.11.
Every subhyperalgebra of mT (Σ , , 𝜅 ) is cdf -generated by 𝐺 ( ) .Proof. Let = ( 𝐴, { 𝜎 } Σ ) be a subhyperalgebra of mT (Σ , , 𝜅 ) , let = ( 𝐵, { 𝜎 } Σ ) be any Σ -hyperalgebra, let 𝑓 ∶ 𝐺 ( ) → 𝐵 be a function and 𝐶 a collection of choices from to . We define 𝑓 𝐶 ∶ → by induction on ⟨ 𝐺 ( ) ⟩ 𝑚 :1. if 𝑎 ∈ ⟨ 𝐺 ( ) ⟩ and 𝑎 ∈ 𝐺 ( ) , we define 𝑓 𝐶 ( 𝑎 ) = 𝑓 ( 𝑎 ) ;2. if 𝑎 ∈ ⟨ 𝐺 ( ) ⟩ and 𝑎 ∈ 𝜎 , for a 𝜎 ∈ Σ , we define 𝑓 𝐶 ( 𝑎 ) = 𝐶𝜎 ( 𝑎 ) ;3. if 𝑓 𝐶 is defined for all elements of ⟨ 𝐺 ( ) ⟩ 𝑚 , 𝑎 , … , 𝑎 𝑛 ∈ ⟨ 𝐺 ( ) ⟩ 𝑚 and 𝜎 ∈ Σ 𝑛 , for every element 𝑎 ∈ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) we define 𝑓 𝐶 ( 𝑎 ) = 𝐶𝜎 𝑓 𝐶 ( 𝑎 ) , … ,𝑓 𝐶 ( 𝑎 𝑛 ) 𝑎 , … ,𝑎 𝑛 ( 𝑎 ) . First, we must prove that 𝑓 𝐶 is well defined. There are two possibly problematic cases to consider for anelement 𝑎 ∈ 𝐴 : the one in which 𝑎 ∈ 𝐺 ( ) and there are 𝜎 ∈ Σ 𝑛 and 𝑎 , … , 𝑎 𝑛 ∈ 𝐴 for which 𝑎 ∈ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) ,corresponding to 𝑎 falling simultaneously in the cases (1) and (2) , or (1) and (3) of the definition; and the one wherethere are 𝜎 ∈ Σ 𝑛 , 𝜃 ∈ Σ 𝑚 , 𝑎 , … , 𝑎 𝑛 ∈ 𝐴 and 𝑏 , … , 𝑏 𝑚 ∈ 𝐴 such that 𝑎 ∈ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) and 𝑎 ∈ 𝜃 ( 𝑏 , … , 𝑏 𝑚 ) ,corresponding to the cases (2) and (3) , (2) and (2) , or (3) and (3) occurring simultaneously.The first case is not possible, since 𝐺 ( ) ⊆ 𝐴 ⧵ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) for every 𝜎 ∈ Σ 𝑛 and 𝑎 , … , 𝑎 𝑛 ∈ 𝐴 ; in thesecond case, we find that 𝑎 ∈ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) ∩ 𝜃 ( 𝑏 , … , 𝑏 𝑚 ) ⊆ 𝜎 mT (Σ , ,𝜅 ) ( 𝑎 , … , 𝑎 𝑛 ) ∩ 𝜃 mT (Σ , ,𝜅 ) ( 𝑏 , … , 𝑏 𝑚 ) , so 𝑛 = 𝑚 , 𝜎 = 𝜃 and 𝑎 = 𝑏 , … , 𝑎 𝑛 = 𝑏 𝑚 , and therefore 𝑓 𝐶 ( 𝑎 ) is well-defined.Second, we must prove that 𝑓 𝐶 is defined over all of 𝐴 : that is simple, for 𝑓 𝐶 is defined over all of ⟨ 𝐺 ( ) ⟩ andwe established in Lemma 3.10 that 𝐴 = ⟨ 𝐺 ( ) ⟩ .So 𝑓 𝐶 ∶ 𝐴 → 𝐵 is a well-defined function: it remains to be shown that it is an homomorphism; given 𝜎 ∈ Σ 𝑛 and 𝑎 , … , 𝑎 𝑛 , we see that 𝑓 𝐶 ( 𝜎 ( 𝑎 , … , 𝑎 𝑛 )) = { 𝐶𝜎 𝑓 𝐶 ( 𝑎 ) , … ,𝑓 𝐶 ( 𝑎 𝑛 ) 𝑎 , … ,𝑎 𝑛 ( 𝑎 ) ∶ 𝑎 ∈ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) } ⊆ 𝜎 ( 𝑓 𝐶 ( 𝑎 ) , … , 𝑓 𝐶 ( 𝑎 𝑛 )) , while we also have that 𝑓 𝐶 clearly extends both 𝑓 and all 𝐶𝜎 𝑓 𝐶 ( 𝑎 , … ,𝑓 𝐶 ( 𝑎 𝑛 ) 𝑎 , … ,𝑎 𝑛 .To finish the proof, suppose 𝑔 ∶ → is another homomorphism extending both 𝑓 and all 𝐶𝜎 𝑔 ( 𝑎 , … ,𝑔 ( 𝑎 𝑛 ) 𝑎 , … ,𝑎 𝑛 :we will prove that 𝑔 = 𝑓 𝐶 again by induction on the 𝑚 of ⟨ 𝐺 ( ) ⟩ 𝑚 . For 𝑚 = 0 , an element 𝑎 ∈ ⟨ 𝐺 ( ) ⟩ is eitherin 𝐺 ( ) , when we have 𝑔 ( 𝑎 ) = 𝑓 ( 𝑎 ) = 𝑓 𝐶 ( 𝑎 ) , or in 𝜎 for a 𝜎 ∈ Σ , when 𝑔 ( 𝑎 ) = 𝐶𝜎 ( 𝑎 ) = 𝑓 𝐶 ( 𝑎 ) .Suppose 𝑔 is equal to 𝑓 𝐶 in ⟨ 𝐺 ( ) ⟩ 𝑚 and take an 𝑎 ∈ ⟨ 𝐺 ( ) ⟩ 𝑚 +1 ⧵ ⟨ 𝐺 ( ) ⟩ 𝑚 : we have that there exist 𝜎 ∈ Σ 𝑛 and 𝑎 , … , 𝑎 𝑛 ∈ ⟨ 𝐺 ( ) ⟩ 𝑚 such that 𝑎 ∈ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) , and then 𝑔 ( 𝑎 ) = 𝐶𝜎 𝑔 ( 𝑎 ) , … ,𝑔 ( 𝑎 𝑛 ) 𝑎 , … ,𝑎 𝑛 ( 𝑎 ) = 𝐶𝜎 𝑓 𝐶 ( 𝑎 ) , … ,𝑓 𝐶 ( 𝑎 𝑛 ) 𝑎 , … ,𝑎 𝑛 ( 𝑎 ) = 𝑓 𝐶 ( 𝑎 ) , proving that 𝑔 = 𝑓 𝐶 and that, in fact, 𝑓 𝐶 is unique. This means that is cdf -generated by 𝐺 ( ) .bsolutely Free Hyperalgebras 7The proof of the following lemma may be found in section 2 of [4]. Lemma 3.12. If = ( 𝐴, { 𝜎 } 𝜎 ∈Σ ) and = ( 𝐵, { 𝜎 } 𝜎 ∈Σ ) are Σ -hyperalgebras and 𝑓 ∶ → is an homomor-phism, = ( 𝑓 ( 𝐴 ) , { 𝜎 } 𝜎 ∈Σ ) such that 𝜎 ( 𝑐 , … , 𝑐 𝑛 ) = ⋃ { 𝑓 ( 𝜎 ( 𝑎 , … , 𝑎 𝑛 )) ∶ 𝑓 ( 𝑎 ) = 𝑐 , … , 𝑓 ( 𝑎 𝑛 ) = 𝑐 𝑛 } is a Σ -subhyperalgebra of , while 𝑓 ∶ → is an epimorphism. The Σ -hyperalgebra is known as the directimage of trough 𝑓 . Theorem 3.13.
If the hyperalgebra = ( 𝐴, { 𝜎 } 𝜎 ∈Σ ) over Σ is cdf -generated by 𝑋 , then is isomorphic to asubhyperalgebra of mT (Σ , 𝑋, | 𝐴 | ) containing 𝑋 .Proof. Take 𝑓 ∶ 𝑋 → 𝑇 (Σ | 𝐴 | , 𝑋 ) to be the inclusion (such that 𝑓 ( 𝑥 ) = 𝑥 ), and take a collection of choices 𝐶 suchthat, for 𝜎 ∈ Σ 𝑛 , 𝑎 , … , 𝑎 𝑛 ∈ 𝐴 and 𝛼 , … , 𝛼 𝑛 ∈ 𝑇 (Σ | 𝐴 | , 𝑋 ) , 𝐶𝜎 𝛼 , … ,𝛼 𝑛 𝑎 , … ,𝑎 𝑛 ∶ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) → 𝜎 mT (Σ ,𝑋, | 𝐴 | ) ( 𝛼 , … , 𝛼 𝑛 ) is an injective function. Such collection of choices exist since 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) ⊆ 𝐴 and 𝜎 mT (Σ ,𝑋, | 𝐴 | ) ( 𝛼 , … , 𝛼 𝑛 ) isof cardinality | 𝐴 | . Now, since is cdf -generated by 𝑋 , there exists an homomorphism 𝑓 𝐶 ∶ → mT (Σ , 𝑋, | 𝐴 | ) extending 𝑓 and each 𝐶𝜎 𝑓 𝐶 ( 𝑎 ) , … ,𝑓 𝐶 ( 𝑎 𝑛 ) 𝑎 , … ,𝑎 𝑛 .Let = ( 𝑓 𝐶 ( 𝐴 ) , { 𝜎 } 𝜎 ∈Σ ) be the direct image of trough 𝑓 𝐶 , so that 𝑓 𝐶 ∶ → is an epimorphism, whatis possible given 3.12: notice too that 𝑋 = 𝑋 ∩ 𝑓 𝐶 ( 𝐴 ) = 𝐺 ( mT (Σ , 𝑋, | 𝐴 | )) ∩ 𝑓 𝐶 ( 𝐴 ) ⊆ 𝐺 ( ) because is a subhyperalgebra of mT (Σ , 𝑋, | 𝐴 | ) . Now, take any 𝑔 ∶ 𝐺 ( ) → 𝐴 such that 𝑔 ( 𝑥 ) = 𝑥 , for every 𝑥 ∈ 𝑋 , and a collection of choices 𝐷 from to such that, for any 𝜎 ∈ Σ 𝑛 , 𝑏 , … , 𝑏 𝑛 ∈ 𝑓 𝐶 ( 𝐴 ) and 𝑎 , … , 𝑎 𝑛 ∈ 𝐴 ,the function 𝐷𝜎 𝑎 , … ,𝑎 𝑛 𝑏 , … ,𝑏 𝑛 ∶ 𝜎 ( 𝑏 , … , 𝑏 𝑛 ) → 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) satisfies the following: if 𝑎 ∈ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) is such that 𝐶𝜎 𝑏 , … ,𝑏 𝑛 𝑎 , … ,𝑎 𝑛 ( 𝑎 ) ∈ 𝜎 ( 𝑏 , … , 𝑏 𝑛 ) then 𝐷𝜎 𝑎 , … ,𝑎 𝑛 𝑏 , … ,𝑏 𝑛 ( 𝐶𝜎 𝑏 , … ,𝑏 𝑛 𝑎 , … ,𝑎 𝑛 ( 𝑎 )) = 𝑎 . Given that 𝐶𝜎 𝑏 , … ,𝑏 𝑛 𝑎 , … ,𝑎 𝑛 is injective, this condition is well-defined.Since is cdf -generated by 𝐺 ( ) , we know to exist an homomorphism 𝑔 𝐷 ∶ → extending 𝑔 and thefunctions 𝐷𝜎 𝑔 𝐷 ( 𝑏 ) , … ,𝑔 𝐷 ( 𝑏 𝑛 ) 𝑏 , … ,𝑏 𝑛 .Finally, we take 𝑔 𝐷 ◦ 𝑓 𝐶 ∶ → : it extends the injection 𝑖𝑑 = 𝑔 ◦ 𝑓 ∶ 𝑋 → 𝐴 , for which 𝑖𝑑 ( 𝑥 ) = 𝑥 ; it alsoextends the collection of choices 𝐸 defined by 𝐸𝜎 𝑎 ′1 , … ,𝑎 ′ 𝑛 𝑎 , … ,𝑎 𝑛 = 𝐷𝜎 𝑎 ′1 , … ,𝑎 ′ 𝑛 𝑓 𝐶 ( 𝑎 ) , … ,𝑓 𝐶 ( 𝑎 𝑛 ) ◦ 𝐶 𝑓 𝐶 ( 𝑎 ) , … ,𝑓 𝐶 ( 𝑎 𝑛 ) 𝑎 , … ,𝑎 𝑛 ∶ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) → 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) , for 𝜎 ∈ Σ 𝑛 and 𝑎 , … 𝑎 𝑛 , 𝑎 ′1 , … , 𝑎 ′ 𝑛 ∈ 𝐴 . This way, 𝐸𝜎 𝑎 , … ,𝑎 𝑛 𝑎 , … ,𝑎 𝑛 is the identity on 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) : indeed, for any 𝑎 ∈ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) , 𝐶𝜎 𝑓 𝐶 ( 𝑎 ) , … ,𝑓 𝐶 ( 𝑎 𝑛 ) 𝑎 , … ,𝑎 𝑛 ( 𝑎 ) = 𝑓 𝐶 ( 𝑎 ) by definition of 𝑓 𝐶 , and, given that 𝑓 𝐶 is an homomorphism, 𝑓 𝐶 ( 𝑎 ) ∈ 𝜎 ( 𝑓 𝐶 ( 𝑎 , … , 𝑓 𝐶 ( 𝑎 𝑛 )) , meaning that 𝐶𝜎 𝑓 𝐶 ( 𝑎 ) , … ,𝑓 𝐶 ( 𝑎 𝑛 ) 𝑎 , … ,𝑎 𝑛 ( 𝑎 ) ∈ 𝜎 ( 𝑓 𝐶 ( 𝑎 ) , … , 𝑓 𝐶 ( 𝑎 𝑛 )) . Then 𝐸𝜎 𝑎 , … ,𝑎 𝑛 𝑎 , … ,𝑎 𝑛 ( 𝑎 ) = 𝐷𝜎 𝑎 , … ,𝑎 𝑛 𝑓 𝐶 ( 𝑎 ) , … ,𝑓 𝐶 ( 𝑎 𝑛 ) ( 𝐶 𝑓 𝐶 ( 𝑎 ) , … ,𝑓 𝐶 ( 𝑎 𝑛 ) 𝑎 , … ,𝑎 𝑛 ( 𝑎 )) = 𝑎 by definition of 𝐷 .But notice that the identical homomorphism ∶ → also extends both 𝑖𝑑 and 𝐸 and, given the unicityof such extensions on the definition of being cdf -generated, we obtain that = 𝑔 𝐷 ◦ 𝑓 𝐶 . Of course, the fact that 𝑓 𝐶 ∶ → has a left inverse implies that is injective, and by definition of it is also surjective, meaning it is abijective function. Moreover, 𝑔 𝐷 is the inverse function of 𝑓 𝐶 . Finally, for 𝜎 ∈ Σ 𝑛 and 𝑎 , … , 𝑎 𝑛 ∈ 𝐴 , 𝑓 𝐶 ( 𝜎 ( 𝑎 , … , 𝑎 𝑛 )) ⊆ 𝜎 ( 𝑓 𝐶 ( 𝑎 ) , … , 𝑓 𝐶 ( 𝑎 𝑛 )) , since 𝑓 𝐶 is an homomorphism; however, given 𝑔 𝐷 is also an homomorphism. 𝑔 𝐷 ( 𝜎 ( 𝑓 𝐶 ( 𝑎 ) , … , 𝑓 𝐶 ( 𝑎 𝑛 ))) ⊆ 𝜎 ( 𝑔 𝐷 ◦ 𝑓 𝐶 ( 𝑎 ) , … , 𝑔 𝐷 ◦ 𝑓 𝐶 ( 𝑎 𝑛 )) = 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) , and by applying 𝑓 𝐶 to both sides, one obtains 𝜎 ( 𝑓 𝐶 ( 𝑎 ) , … , 𝑓 𝐶 ( 𝑎 𝑛 )) = 𝑓 𝐶 ( 𝑔 𝐷 ( 𝜎 ( 𝑓 𝐶 ( 𝑎 ) , … , 𝑓 𝐶 ( 𝑎 𝑛 )))) ⊆ 𝑓 𝐶 ( 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) , what proves that 𝑓 𝐶 is a bijective full homomorphism, that is, an isomorphism.bsolutely Free Hyperalgebras 8Notice that, from the proof above, we can see that in fact if = ( 𝐴, { 𝜎 } 𝜎 ∈Σ ) is cdf -generated by 𝑋 , then is isomorphic to a subhyperalgebra of mT (Σ , 𝑋, 𝑀 ( )) , where 𝑀 ( ) = max {| 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) | ∶ 𝑛 ∈ ℕ , 𝜎 ∈ Σ 𝑛 , 𝑎 , … , 𝑎 𝑛 ∈ 𝐴 } . It is clear that 𝑀 ( ) = 𝜅 for the hyperalgebra mT (Σ , , 𝜅 ) . The value 𝑀 ( ) has been already regarded in theliterature as an important aspect of hyperalgebras, see [6] (observe, however, that their definition of homomorphismis profoundly different from ours).Notice, furthermore, that written in classical therms, the previous Theorems 3.11 and 3.13 state somethingquite well known: an algebra is absolutely free if, and only if, it is isomorphic to some algebra of terms over thesame signature. Corollary 3.14.
Every cdf -generated hyperalgebra is generated by its ground 𝐺 ( ) .Proof. Since every cdf -generated hyperalgebra is isomorphic to a subhyperalgebra of some mT (Σ , 𝑋, 𝜅 ) , from3.13, and every subhyperalgebra of mT (Σ , 𝑋, 𝜅 ) is generated by its ground, the result follows. Corollary 3.15.
Every cdf -generated hyperalgebra is cdf -generated by its ground 𝐺 ( ) . Definition 3.16. A Σ -hyperalgebra = ( 𝐴, { 𝜎 } 𝜎 ∈Σ ) is said to be disconnected if, for every 𝜎 ∈ Σ 𝑛 , 𝜃 ∈ Σ 𝑚 , 𝑎 , … , 𝑎 𝑛 , 𝑏 , … , 𝑏 𝑚 ∈ 𝐴 , 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) ∩ 𝜃 ( 𝑏 , … , 𝑏 𝑚 ) ≠ ∅ implies that 𝑛 = 𝑚 , 𝜎 = 𝜃 and 𝑎 = 𝑏 , … , 𝑎 𝑛 = 𝑏 𝑚 . Example 3.17. T (Σ , ) is disconnected. Example 3.18.
All directed forests of height 𝜔 , when considered as Σ 𝑠 -hyperalgebras, are disconnected, giventhat no two arrows point to the same element. It is clear that if is a subalgebra of and is disconnected, then is also disconnected, since if 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) ∩ 𝜃 ( 𝑏 , … , 𝑏 𝑚 ) ≠ ∅ , for 𝑎 , … , 𝑎 𝑛 , 𝑏 , … , 𝑏 𝑚 ∈ 𝐵 , given that 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) ⊆ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) and 𝜃 ( 𝑏 , … , 𝑏 𝑚 ) ⊆ 𝜃 ( 𝑏 , … , 𝑏 𝑚 ) , we find that 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) ∩ 𝜃 ( 𝑏 , … , 𝑏 𝑚 ) ≠ ∅ and therefore 𝑛 = 𝑚 , 𝜎 = 𝜃 and 𝑎 = 𝑏 , … , 𝑎 𝑛 = 𝑏 𝑚 .We noticed before that mT (Σ , , 𝜅 ) is disconnected, and by the theorem we obtain that every cdf − generatedalgebra is disconnected. This also means something deeper: being disconnected is, in a way, a measure of howfree of identities a hyperalgebra is. After all, having no two hyperoperations to coincide, on any elements, is astrong indicative that the hyperalgebra does not satisfy any identities. The fact that our hyperalgebras of terms areall disconnected is indicative that they are well-deserving of the “absolutely free” title. Now, we continue to look at other possible characterizations of our hyperalgebras of terms that could lead to futuregeneralizations of relatively free algebras. One sees that algebras of terms do not have identities, what wouldpartially correspond in our study to the concept of being disconnected; but, what is possibly more representativeof our intuition behind terms, is that one starts by defining them from elements that are as simple as possible(variables), and continue indefinitely. The concept of indecomposable element, here, is replaced by that of theground, so one would expect that being generated by it plays some sort of role in what we have defined so far.
Lemma 3.19. If is cdf -generated by 𝑋 , then 𝑋 ⊆ 𝐺 ( ) .Proof. If is cdf -generated by 𝑋 , then is isomorphic to a subhyperalgebra of mT (Σ , 𝑋, | 𝐴 | ) containing 𝑋 ,from 3.13: let us assume that is equal to this subhyperalgebra, without loss of generality.Then we have 𝑋 = 𝐺 ( mT (Σ , 𝑋, | 𝐴 | )) ∩ 𝐴 ⊆ 𝐺 ( ) . Lemma 3.20. If is cdf -generated by both 𝑋 and 𝑌 , with 𝑋 ⊆ 𝑌 , then 𝑋 = 𝑌 .Proof. Suppose 𝑋 ≠ 𝑌 and let 𝑦 ∈ 𝑌 ⧵ 𝑋 : take a Σ -hyperalgebra over the same signature as such that | 𝐵 | ≥ , and a collection of choices 𝐶 from to .Take also two functions 𝑔, ℎ ∶ 𝑌 → 𝐵 such that 𝑔 | 𝑋 = ℎ | 𝑋 = 𝑓 and 𝑔 ( 𝑦 ) ≠ ℎ ( 𝑦 ) , which is possible since | 𝐵 | ≥ : given that is cdf -generated by 𝑌 , there exist unique homomorphisms 𝑔 𝐶 and ℎ 𝐶 extending both,respectively, 𝑔 and 𝐶 and ℎ and 𝐶 .However, 𝑔 𝐶 and ℎ 𝐶 extend both 𝑔 | 𝑋 ∶ 𝑋 → 𝐵 and 𝐶 , and since is freely generated by 𝑋 , we find that 𝑔 𝐶 = ℎ 𝐶 . This is not possible, since 𝑔 𝐶 ( 𝑦 ) ≠ ℎ 𝐶 ( 𝑦 ) , what must implies that 𝑌 ⧵ 𝑋 = ∅ and therefore 𝑋 = 𝑌 .bsolutely Free Hyperalgebras 9 Theorem 3.21.
Every cdf -generated hyperalgebra is uniquely cdf -generated by its ground.Proof. From 3.14, is cdf -generated by 𝐺 ( ) , and from 3.19, if is also cdf -generated by 𝑋 , then 𝑋 ⊆ 𝐺 ( ) .By 3.20, this implies that 𝑋 = 𝐺 ( ) .We have proved so far that, if is cdf -generated, then is generated by its ground and disconnected. Wewould like to prove that this is enough to characterize a cdf -generated hyperalgebra: that is, if is generated byits ground and disconnected, then it is cdf -generated, exactly by its ground.The idea is similar to the one we used to prove that all subhyperalgebras of mT (Σ , , 𝜅 ) are cdf -generated:take a hyperalgebra that is both generated by its ground 𝐺 ( ) , which we will denote by 𝑋 , and disconnected,and fix a hyperalgebra over the same signature, a function 𝑓 ∶ 𝑋 → 𝐵 and a collection of choices 𝐶 from to . We define a function 𝑓 𝐶 ∶ 𝐴 → 𝐵 using induction on the ⟨ 𝑋 ⟩ 𝑛 : for 𝑛 = 0 , either we have an element 𝑥 ∈ 𝑋 ,when we define 𝑓 𝐶 ( 𝑥 ) = 𝑓 ( 𝑥 ) , or we have an 𝑎 ∈ 𝜎 for a 𝜎 ∈ Σ , when we define 𝑓 𝐶 ( 𝑎 ) = 𝐶𝜎 ( 𝑎 ) . Notice how,up to this point, there are no contradictions on the definition, given an element cannot belong both to 𝑋 and to a 𝜎 , since 𝑋 = 𝐺 ( ) .Suppose we have successfully defined 𝑓 𝐶 on ⟨ 𝑋 ⟩ 𝑚 and take an 𝑎 ∈ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) for 𝑎 , … , 𝑎 𝑛 ∈ ⟨ 𝑋 ⟩ 𝑚 . Wethen define 𝑓 𝐶 ( 𝑎 ) = 𝐶𝜎 𝑓 𝐶 ( 𝑎 ) , … ,𝑓 𝐶 ( 𝑎 𝑛 ) 𝑎 , … ,𝑎 𝑛 ( 𝑎 ) . Again the function remains well-defined: 𝑎 cannot belong to 𝑋 , since 𝑋 = 𝐺 ( ) , and cannot belong to a 𝜃 ( 𝑏 , … , 𝑏 𝑝 ) unless 𝑝 = 𝑛 , 𝜃 = 𝜎 and 𝑏 = 𝑎 , … , 𝑏 𝑝 = 𝑎 𝑛 , since is disconnected.Clearly 𝑓 𝐶 is an homomorphism, since the image of 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) under 𝑓 𝐶 is contained in 𝜎 ( 𝑓 𝐶 ( 𝑎 ) , … , 𝑓 𝐶 ( 𝑎 𝑛 )) , and 𝑓 𝐶 extends both 𝑓 and 𝐶 . Lemma 3.22.
If a hyperalgebra is both generated by its ground 𝑋 and disconnected, is cdf -generated by 𝑋 .Proof. It remains for us to show that 𝑓 𝐶 , as defined above, is the only homomorphism extending 𝑓 and 𝐶 . Suppose 𝑔 is another such homomorphism and we shall proceed yet again by induction.On ⟨ 𝑋 ⟩ , we have that 𝑓 𝐶 ( 𝑥 ) = 𝑓 ( 𝑥 ) = 𝑔 ( 𝑥 ) for all 𝑥 ∈ 𝑋 ; and for 𝑎 ∈ 𝜎 and 𝜎 ∈ Σ we have that 𝑓 𝐶 ( 𝑎 ) = 𝐶𝜎 ( 𝑎 ) = 𝑔 ( 𝑎 ) , hence 𝑓 𝐶 and 𝑔 coincide on ⟨ 𝑋 ⟩ . Suppose that 𝑓 𝐶 and 𝑔 are equal on ⟨ 𝑋 ⟩ 𝑚 and take 𝑎 ∈ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) for 𝑎 , … , 𝑎 𝑛 ∈ ⟨ 𝑋 ⟩ 𝑚 : we have by induction hypothesis that 𝑓 𝐶 ( 𝑎 ) = 𝐶𝜎 𝑓 𝐶 ( 𝑎 ) , … ,𝑓 𝐶 ( 𝑎 𝑛 ) 𝑎 , … ,𝑎 𝑛 ( 𝑎 ) = 𝐶𝜎 𝑔 ( 𝑎 ) , … ,𝑔 ( 𝑎 𝑛 ) 𝑎 , … ,𝑎 𝑛 ( 𝑎 ) = 𝑔 ( 𝑎 ) , which concludes our proof. Theorem 3.23.
A hyperalgebra is cdf -generated if, and only if, is generated by its ground and disconnected. It is important to analyze the differences between the several concepts involved with examples: are there hy-peralgebras that are disconnected but not generated by their grounds? Are there hyperalgebras that are generatedby their grounds but not disconnected? If not, does being generated by its ground imply being disconnected orvice-versa? We show below that this is not the case, for we provide examples answering positively both previousquestions.
Example 3.24.
Take the signature Σ 𝑠 from Example 2.4. Consider the Σ 𝑠 − hyperalgebra = ({−1 , , { 𝑠 }) suchthat 𝑠 (−1) = {1} and 𝑠 (1) = {−1} (that is, 𝑠 ( 𝑥 ) = {− 𝑥 } ).We state that is disconnected, but not generated by its ground. is clearly disconnected since 𝑠 (−1)∩ 𝑠 (1) =∅ ; now, 𝐵 ( ) = 𝑠 (−1) ∪ 𝑠 (1) = {−1 , , and so 𝐺 ( ) = ∅ .Since Σ 𝑠 has no -ary operators and 𝐺 ( ) = ∅ , it follows that ⟨ 𝐺 ( ) ⟩ = ∅ and therefore ⟨ 𝐺 ( ) ⟩ 𝑛 = ∅ forevery 𝑛 ∈ ℕ , meaning that 𝐺 ( ) does not generate . Example 3.25.
Take again the signature Σ 𝑠 with a single unary operator, from Example 2.4. Consider the Σ 𝑠 -hyperalgebra = ({0 , , { 𝑠 }) such that 𝑠 (0) = {1} and 𝑠 (1) = {1} (that is, 𝑠 ( 𝑥 ) = {1} ).Then is clearly not disconnected, since 𝑠 (0) ∩ 𝑠 (1) = {1} , yet is generated by its ground: 𝐵 ( ) = {1} and so 𝐺 ( ) = {0} , and we see that ⟨ 𝐺 ( ) ⟩ is already {0 , . bsolutely Free Hyperalgebras 10 −1 1 𝑠 𝑠 The Σ 𝑠 -hyperalgebra 𝑠 𝑠 The Σ 𝑠 -hyperalgebra We give another alternative approach to being cdf -generated, that is being disconnected and having a strong basis,in a sense we now define. Our main motivation, when considering the notion of a strong basis, was to be able toweaken that very condition: remember, absolutely free algebras are easier to define than the relatively free ones,so that is why we start with absolutely free hyperalgebras. Nonetheless, we still wish to be able in the future todefine what would be a relatively free hyperalgebra, whose classical counterparts, on many environments as thatof vector spaces, have basis defined as minimal generating sets.
Definition 3.26.
We say
𝐵 ⊆ 𝐴 is a strong basis of the Σ -hyperalgebra = ( 𝐴, { 𝜎 } 𝜎 ∈Σ ) if it is the minimum ofthe set = { 𝑆 ⊆ 𝐴 ∶ ⟨ 𝑆 ⟩ = 𝐴 } when ordered by inclusion. Example 3.27.
The set of variables is a strong basis of T (Σ , ) . Example 3.28.
The set of elements without predecessor of a directed forest of height 𝜔 is a strong basis of theforest, when considered as a Σ 𝑠 -hyperalgebra. Lemma 3.29.
For every subset 𝑆 of the universe of a Σ -hyperalgebra , 𝐺 ( ) ∩ ⟨ 𝑆 ⟩ ⊆ 𝑆 .Proof. Suppose 𝑥 ∈ 𝐺 ( )∩ ⟨ 𝑆 ⟩ : if 𝑥 ∉ 𝑆 , we will show that 𝑥 cannot be in ⟨ 𝑆 ⟩ , which contradicts our assumption.Indeed, if 𝑥 ∉ 𝑆 then 𝑥 ∉ ⟨ 𝑆 ⟩ = 𝑆 ∪ ⋃ 𝜎 ∈Σ 𝜎 , since 𝑥 ∉ 𝑆 , and 𝑥 ∈ 𝐺 ( ) implies that 𝑥 ∈ 𝐴 ⧵ 𝐵 ( ) ⊆ 𝐴 ⧵ ⋃ 𝜎 ∈Σ 𝜎 . Now, for induction hypothesis, suppose that 𝑥 ∉ ⟨ 𝑆 ⟩ 𝑚 ; then 𝑥 ∉ ⟨ 𝑆 ⟩ 𝑚 +1 = ⟨ 𝑆 ⟩ 𝑚 ∪ ⋃ { 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) ∶ 𝑛 ∈ ℕ , 𝜎 ∈ Σ 𝑛 , 𝑎 , … , 𝑎 𝑛 ∈ ⟨ 𝑆 ⟩ 𝑚 } since 𝑥 ∉ ⟨ 𝑆 ⟩ 𝑚 , and 𝑥 ∈ 𝐺 ( ) implies that 𝑥 ∈ 𝐴 ⧵ 𝐵 ( ) ⊆ 𝐴 ⧵ ⋃ { 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) ∶ 𝑛 ∈ ℕ , 𝜎 ∈ Σ 𝑛 , 𝑎 , … , 𝑎 𝑛 ∈ ⟨ 𝑆 ⟩ 𝑚 } . Theorem 3.30.
If the Σ -hyperalgebra has a strong basis 𝐵 , 𝐺 ( ) ⊆ 𝐵 .Proof. By 3.29, 𝐺 ( ) = 𝐺 ( ) ∩ 𝐴 = 𝐺 ( ) ∩ ⟨ 𝐵 ⟩ ⊆ 𝐵 . Definition 3.31. If 𝐵 is a strong basis of a disconnected Σ -hyperalgebra , we define the 𝐵 -order of an element 𝑎 ∈ 𝐴 as the natural number 𝑜 𝐵 ( 𝑎 ) = min { 𝑘 ∈ ℕ ∶ 𝑎 ∈ ⟨ 𝐵 ⟩ 𝑘 } . This is a clear generalization of the order, or complexity, of a term: in fact, the order of a term in 𝑇 (Σ , ) isexactly its -order.It is clear that, if 𝑎 ∈ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) and 𝑜 𝐵 ( 𝑎 ) ≥ , then 𝑜 𝐵 ( 𝑎 ) , … , 𝑜 𝑏 ( 𝑎 𝑛 ) < 𝑜 𝐵 ( 𝑎 ) : in fact, suppose 𝑚 + 1 = 𝑜 𝐵 ( 𝑎 ) , implying 𝑎 ∈ ⟨ 𝐵 ⟩ 𝑚 +1 = ⟨ 𝐵 ⟩ 𝑚 ∪ ⋃ { 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) ∶ 𝑛 ∈ ℕ , 𝜎 ∈ Σ 𝑛 , 𝑎 , … , 𝑎 𝑛 ∈ ⟨ 𝐵 ⟩ 𝑚 } ; bsolutely Free Hyperalgebras 11since 𝑚 = min{ 𝑘 ∈ ℕ ∶ 𝑎 ∈ ⟨ 𝐵 ⟩ 𝑘 } , we have that 𝑎 ∉ ⟨ 𝐵 ⟩ 𝑚 and therefore 𝑎 ∈ ⋃ { 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) ∶ 𝑛 ∈ ℕ , 𝜎 ∈ Σ 𝑛 , 𝑎 , … , 𝑎 𝑛 ∈ ⟨ 𝐵 ⟩ 𝑚 } . Finally, we obtain that there exist 𝑝 ∈ ℕ , 𝜃 ∈ Σ 𝑝 and 𝑏 , … , 𝑏 𝑝 ∈ ⟨ 𝐵 ⟩ 𝑚 such that 𝑎 ∈ 𝜃 ( 𝑏 , … , 𝑏 𝑝 ) . Since 𝑎 ∈ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) , this implies that 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) ∩ 𝜃 ( 𝑏 , … , 𝑏 𝑝 ) ≠ ∅ , and therefore 𝑝 = 𝑛 , 𝜃 = 𝜎 and 𝑏 = 𝑎 ,. . . , 𝑏 𝑝 = 𝑎 𝑛 , so that 𝑜 𝐵 ( 𝑎 ) , … , 𝑜 𝐵 ( 𝑎 𝑛 ) ≤ 𝑚 .But what if 𝑎 ∈ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) , for 𝑛 > , and 𝑜 𝐵 ( 𝑎 ) = 0 , implying 𝑎 ∈ 𝐵 ? We state this case cannot occur,for if it does, 𝐵 ∗ = ( 𝐵 ∪ { 𝑎 , … , 𝑎 𝑛 } ) ⧵ { 𝑎 } generates 𝐴 , while clearly not containing 𝐵 , even in the case where 𝑛 = 0 . We have that 𝑎 ∈ ⟨ 𝐵 ∗ ⟩ , since 𝑎 , … , 𝑎 𝑛 ∈ 𝐵 ∗ and 𝑎 ∈ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) , and given that 𝐵 ⧵ { 𝑎 } ⊆ 𝐵 ∗ , it follows that 𝐵 ⊆ ⟨ 𝐵 ∗ ⟩ , and so ⟨ 𝐵 ⟩ ⊆ ⟨ 𝐵 ∗ ⟩ .It is then true that, for every 𝑚 ∈ ℕ , ⟨ 𝐵 ⟩ 𝑚 ⊆ ⟨ 𝐵 ∗ ⟩ 𝑚 +1 : if this is true for 𝑚 , let 𝑏 ∈ ⟨ 𝐵 ⟩ 𝑚 +1 , and theneither 𝑏 ∈ ⟨ 𝐵 ⟩ 𝑚 , so that 𝑏 ∈ ⟨ 𝐵 ∗ ⟩ 𝑚 +1 ⊆ ⟨ 𝐵 ∗ ⟩ 𝑚 +2 , or there exist 𝜃 ∈ Σ 𝑝 and 𝑏 , … , 𝑏 𝑝 ∈ ⟨ 𝐵 ⟩ 𝑚 such that 𝑏 ∈ 𝜃 ( 𝑏 , … , 𝑏 𝑝 ) . In this case, since ⟨ 𝐵 ⟩ 𝑚 ⊆ ⟨ 𝐵 ∗ ⟩ 𝑚 +1 , we have that 𝑏 ∈ 𝜃 ( 𝑏 , … , 𝑏 𝑝 ) ⊆ ⋃ { 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) ∶ 𝑛 ∈ ℕ , 𝜎 ∈ Σ 𝑛 , 𝑎 , … , 𝑎 𝑛 ∈ ⟨ 𝐵 ∗ ⟩ 𝑚 +1 } ⊆ ⟨ 𝐵 ∗ ⟩ 𝑚 +2 , so once again 𝑏 ∈ ⟨ 𝐵 ∗ ⟩ 𝑚 +2 . Since ⟨ 𝐵 ⟩ = ⋃ 𝑚 ∈ ℕ ⟨ 𝐵 ⟩ 𝑚 equals 𝐴 , we have that ⟨ 𝐵 ∗ ⟩ also equals 𝐴 , as we previouslystated. This is absurd, since 𝐵 is the minimum of { 𝑆 ⊆ 𝐴 ∶ ⟨ 𝑆 ⟩ = 𝐴 } ordered by inclusion and 𝐵 ⊈ 𝐵 ∗ . Theconclusion must be that if 𝑎 ∈ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) , then 𝑜 𝐵 ( 𝑎 ) , … , 𝑜 𝐵 ( 𝑎 𝑛 ) < 𝑜 𝐵 ( 𝑎 ) , regardless of the value of 𝑜 𝐵 ( 𝑎 ) . Lemma 3.32. If is disconnected and has a strong basis 𝐵 , 𝐵 = 𝐺 ( ) and so is generated by its ground.Proof. Suppose 𝑎 ∈ 𝐵 ⧵ 𝐺 ( ) : since 𝑎 is in the build of , there exist 𝜎 ∈ Σ 𝑛 and elements 𝑎 , … , 𝑎 𝑛 ∈ 𝐴 such that 𝑎 ∈ 𝜎 ( 𝑎 , … , 𝑎 𝑛 ) . If 𝑛 > , 𝑜 𝐵 ( 𝑎 ) > 𝑜 𝐵 ( 𝑎 ) ≥ , which contradicts the fact that 𝑎 ∈ 𝐵 and therefore 𝑜 𝐵 ( 𝑎 ) = 0 .If 𝑛 = 0 , it is clear that 𝐵 ∗ = 𝐵 ⧵ { 𝑎 } is a generating set smaller than 𝐵 : generating set because, if 𝑎 ∈ 𝜎 , 𝑎 ∈ ⋃ 𝜎 ∈Σ 𝜎 and therefore 𝐵 ⊆ ⟨ 𝐵 ∗ ⟩ , so that ⟨ 𝐵 ⟩ 𝑚 ⊆ ⟨ 𝐵 ∗ ⟩ 𝑚 +1 . This is also a contradiction, since 𝐵 is a strongbasis. Theorem 3.33. is generated by its ground and disconnected if, and only if, it has a strong basis and it isdisconnected.Proof. We already proved, in 3.32, that if is disconnected and has a strong basis 𝐵 , then it is generated by itsground and disconnected. Reciprocally, if is disconnected and generated by its ground, first of all it is clearlydisconnected.Now, if ⟨ 𝐺 ( ) ⟩ = 𝐴 one has that 𝐺 ( ) ⊆ 𝑆 for every 𝑆 ∈ { 𝑆 ⊆ 𝐴 ∶ ⟨ 𝑆 ⟩ = 𝐴 } , by Lemma 3.29. Therefore,the ground is a strong basis.Once again, we ask ourselves: does being disconnected imply having a strong basis or vice-versa? We showthat this is not the case by providing examples of a hyperalgebra that is disconnected but does not have a strongbasis and one of a hyperalgebra that has a strong basis but is not disconnected. Example 3.34.
Take the signature Σ 𝑠 and the Σ 𝑠 -hyperalgebra from Example 3.24.We know that is disconnected, but we also state that it does not have a strong basis: in fact, we see that theset { 𝑆 ⊆ {−1 ,
1} ∶ ⟨ 𝑆 ⟩ = {−1 , } is exactly { {−1} , {1} , {−1 , } , and this set has no minimum. Example 3.35.
Take, again, the Σ 𝑠 -hyperalgebra from Example 3.25.As we saw before, is not disconnected, but we state that it has a strong basis: 𝐵 = {0} generates and,since {1} does not generate the hyperalgebra, we find that 𝐵 is a minimum generating set. From these two examples, one could hypothesize that being generated by its ground is equivalent to having astrong basis: clearly being generated by its ground implies having a strong basis, that is, the ground; but as weshow in the example below, having a strong basis does not imply being generated by its ground.
Example 3.36.
Take the signature Σ 𝑠 from Example 2.4, and consider the Σ 𝑠 -hyperalgebra = ({−1 , , , { 𝑠 }) such that 𝑠 (0) = {0} , 𝑠 (1) = {1} and 𝑠 (−1) = {1} (that is, 𝑠 ( 𝑥 ) = { 𝖺𝖻𝗌 ( 𝑥 )} , where 𝖺𝖻𝗌 ( 𝑥 ) denotes theabsolute value of 𝑥 ). bsolutely Free Hyperalgebras 12 We have that 𝐺 ( ) = {−1} and that ⟨ {−1} ⟩ = {−1 , , so that is not generated by its ground. But westate that {−1 , is a strong basis: first of all, it clearly generates ; furthermore, the generating sets of areonly {−1 , and {−1 , , , so that {−1 , is in fact the smallest generating set. −1 0 1 𝑠 𝑠 𝑠 The Σ 𝑠 -hyperalgebra The last equivalence to being a subhyperalgebra of mT (Σ , , 𝜅 ) we give depends on the notion of being chainless,which is very graph-theoretical in nature. Think of a tree that ramifies ever downward: one can pick any vertexand proceed, against the arrows, upwards until an element without predecessor is reached. More than that, it is notpossible to find an infinite path, starting in any one vertex, by always going against the arrows: such a path, if itexisted, would be what we shall call a chain. A hyperalgebra without chains is, very naturally, chainless.As it was in the case of strong basis, there isn’t a parallel concept to being chainless among the theory ofuniversal algebra: it seems that this concept is far more natural when dealing with hyperoperations, although it canbe easily applied to algebras if one wishes to do so. A close, although not equivalent, entity are the branches in theformation trees of terms: if allowed to grow infinitely, these would became chains.Given a permutation 𝜏 ∶ {1 , … , 𝑛 } → {1 , … , 𝑛 } in 𝑆 𝑛 , the group of permutations on 𝑛 elements, the action of 𝜏 in an 𝑛 -uple ( 𝑥 , … , 𝑥 𝑛 ) ∈ 𝑋 𝑛 is given by 𝜏 ( 𝑥 , … , 𝑥 𝑛 ) = ( 𝑥 𝜏 (1) , … , 𝑥 𝜏 ( 𝑛 ) ) . Given ≤ 𝑖, 𝑗 ≤ 𝑛 , we define [ 𝑖, 𝑗 ] to be the permutation such that [ 𝑖, 𝑗 ]( 𝑖 ) = 𝑗 , [ 𝑖, 𝑗 ]( 𝑗 ) = 𝑖 and, for 𝑘 ∈ {1 , … , 𝑛 } different from 𝑖 and 𝑗 , [ 𝑖, 𝑗 ]( 𝑘 ) = 𝑘 . Definition 3.37.
Given a Σ -hyperalgebra , a sequence { 𝑎 𝑛 } 𝑛 ∈ ℕ ⊆ 𝐴 is said to be a chain if, for every 𝑛 ∈ ℕ ,there exist a natural number 𝑚 𝑛 ∈ ℕ , a functional symbol 𝜎 𝑛 ∈ Σ 𝑚 𝑛 , a permutation 𝜏 𝑛 ∈ 𝑆 𝑚 𝑛 and elements 𝑎 𝑛 , … , 𝑎 𝑛𝑚 𝑛 −1 ∈ 𝐴 such that 𝑎 𝑛 ∈ 𝜎 𝑛 ( 𝜏 𝑛 ( 𝑎 𝑛 +1 , 𝑎 𝑛 , … , 𝑎 𝑛𝑚 𝑛 −1 )) . A Σ -hyperalgebra is said to be chainless when it has no chains. Example 3.38.
Take a directed forest of height 𝜔 and add to it a loop, that is, choose a vertex 𝑣 and add an arrowfrom 𝑣 to 𝑣 : then { 𝑎 𝑛 } 𝑛 ∈ ℕ , such that 𝑎 𝑛 = 𝑣 for every 𝑛 ∈ ℕ , is a chain. Example 3.39. T (Σ , ) is chainless. Lemma 3.40. If is chainless, then it is generated by its ground.Proof. Suppose that this not hold, so 𝐴 ⧵ ⟨ 𝐺 ( ) ⟩ is not empty, and must therefore contain some element 𝑎 . Wecreate a chain { 𝑎 𝑛 } 𝑛 ∈ ℕ by induction, being the case 𝑛 = 0 already done.So, suppose we have created a finite sequence of elements 𝑎 , … , 𝑎 𝑘 ∈ 𝐴 ⧵ ⟨ 𝐺 ( ) ⟩ such that, for each ≤ 𝑛 < 𝑘 ,there exist a positive integer 𝑚 𝑛 ∈ ℕ ⧵ {0} , a functional symbol 𝜎 𝑛 ∈ Σ 𝑚 𝑛 , a permutation 𝜏 𝑛 ∈ 𝑆 𝑚 𝑛 and elements 𝑎 𝑛 , … , 𝑎 𝑛𝑚 𝑛 −1 ∈ 𝐴 such that 𝑎 𝑛 ∈ 𝜎 𝑛 ( 𝜏 𝑛 ( 𝑎 𝑛 +1 , 𝑎 𝑛 , … , 𝑎 𝑛𝑚 𝑛 −1 )) . Since 𝑎 𝑘 ∈ 𝐴 ⧵ ⟨ 𝐺 ( ) ⟩ , we have that 𝑎 𝑘 is not an element of the ground; so, there must exist 𝑚 𝑘 ∈ ℕ , afunctional symbol 𝜎 𝑘 ∈ Σ 𝑚 𝑘 and elements 𝑏 𝑘 , … , 𝑏 𝑘𝑚 𝑘 ∈ 𝐴 such that 𝑎 𝑘 ∈ 𝜎 𝑘 ( 𝑏 𝑘 , … , 𝑏 𝑘𝑚 𝑘 ) . Now, if all 𝑏 𝑘 , … , 𝑏 𝑘𝑚 𝑘 belonged to ⟨ 𝐺 ( ) ⟩ , so would 𝑎 𝑘 : there must be an element 𝑎 𝑘 +1 ∈ { 𝑏 𝑘 , … , 𝑏 𝑘𝑚 𝑘 } , say 𝑏 𝑘𝑙 ,such that 𝑎 𝑘 +1 ∈ 𝐴 ⧵ ⟨ 𝐺 ( ) ⟩ . We then define 𝑎 𝑘𝑖 as 𝑏 𝑘𝑗 , for 𝑗 = min{ 𝑖 ≤ 𝑝 ≤ 𝑚 𝑘 ∶ 𝑝 ≠ 𝑙 } and 𝑖 ∈ {1 , … , 𝑚 𝑘 − 1} ,and 𝜏 𝑘 = [ 𝑙 − 1 , 𝑙 ] ◦ ⋯ ◦ [1 , , bsolutely Free Hyperalgebras 13and then it is clear that { 𝑎 𝑛 } 𝑛 ∈ ℕ becomes a chain, with the extra condition that { 𝑎 𝑛 } 𝑛 ∈ ℕ ⊆ 𝐴 ⧵ ⟨ 𝐺 ( ) ⟩ . Since is chainless, we reach a contradiction, so we must have instead that 𝐴 ⧵ ⟨ 𝐺 ( ) ⟩ = ∅ , and therefore is generatedby its ground.It becomes clear that a disconnected, chainless hyperalgebra is, by 3.40, disconnected and generated by itsground. We state, that, in fact, the reciprocal also holds, when we arrive at yet another characterization of being asubhyperalgebra of mT (Σ , , 𝜅 ) .So, suppose is disconnected and generated by its ground, and let { 𝑎 𝑛 } 𝑛 ∈ ℕ be a chain in : clearly no 𝑎 𝑛 canbelong to the ground, since 𝑎 𝑛 ∈ 𝜎 𝑛 ( 𝜏 𝑛 ( 𝑎 𝑛 +1 , 𝑎 𝑛 , … , 𝑎 𝑛𝑚 𝑛 −1 )) , and therefore 𝑜 𝐺 ( ) ( 𝑎 𝑛 +1 ) < 𝑜 𝐺 ( ) ( 𝑎 𝑛 ) , that is, the 𝐺 ( ) -order of 𝑎 𝑛 +1 is smaller than the 𝐺 ( ) -order of 𝑎 𝑛 ; wereach a contradiction, since if 𝑜 𝐺 ( ) ( 𝑎 ) = 𝑚 , then 𝑜 𝐺 ( ) ( 𝑎 𝑚 +1 ) < , what is impossible. must then be chainless. Theorem 3.41. is generated by its ground and disconnected if, and only if, it is chainless and disconnected. Finally, the previous results can be summarized as follows:
Theorem 3.42.
Are equivalent:1. is a subhyperalgebra of some mT (Σ , , 𝜅 ) ;2. is cdf -generated;3. is generated by its ground and disconnected;4. has a strong basis and is disconnected;5. is chainless and disconnected. An important point to stress is that, although not all concepts present in the previous theorem have naturalcounterparts in universal algebra, by defining them for algebras, presented as hyperalgebras, we find that all ofthe conditions in the theorem are valid only for algebras of terms. This follows easily from the fact that the only cdf -generated algebras are the algebras of terms themselves.Now, a few examples concerning being chainless, disconnected, having a strong basis and being generated bythe ground will be given.
Example 3.43.
Take the signature Σ 𝑠 from Example 2.4, and consider the Σ 𝑠 − hyperalgebra = ( ℕ ∪{ 𝑎, 𝑏 } , { 𝑠 }) such that 𝑠 ( 𝑛 ) = { 𝑛 + 1} , for 𝑛 ∈ ℕ , and 𝑠 ( 𝑎 ) = 𝑠 ( 𝑏 ) = {0} .We see that is chainless since, given a chain { 𝑎 𝑛 } 𝑛 ∈ ℕ , it must be contained in the build of , that is, ℕ : butthen 𝑎 𝑛 +1 = 𝑎 𝑛 − 1 , what is a contradiction, since there are only a finite number of elements smaller than 𝑎 . Atthe same time, is not disconnected, since 𝑠 ( 𝑎 ) = 𝑠 ( 𝑏 ) . 𝑎 ⋯ 𝑏 𝑠 𝑠 𝑠 𝑠 The Σ 𝑠 -hyperalgebra Example 3.44.
Take the Σ 𝑠 -hyperalgebra from Example 3.24.We know that is disconnected, but we state that it is not chainless: in fact, {(−1) 𝑛 } 𝑛 ∈ ℕ and {(−1) 𝑛 +1 } 𝑛 ∈ ℕ arechains in . As we saw, being chainless implies being generated by its ground and having a strong basis. The reciprocal,however, is not true.
Example 3.45.
Take the Σ 𝑠 -hyperalgebra from Example 3.25.We have already established that has a strong basis and is generated by its ground, {0} , yet it is not chainless: {1} 𝑛 ∈ ℕ is a chain in . bsolutely Free Hyperalgebras 14 Now, we turn to a somewhat folkloric result: the category of hyperalgebras does not have free objects, which canalso be worded as saying that there does not exist absolutely free hyperalgebras, or better yet, that the forgetfulfunctor from this category to
Set does not have a left adjoint. Of course, such a result is stated in various ways,depending on your definition of homomorphism and, perhaps most severely, even on your own definition of hyper-algebra. So, we offer what we believe to be a simplified proof of the result for the category
MAlg (Σ) as we havedefined it.
Definition 4.1. A Σ -hyperalgebra = ( 𝐴, { 𝜎 } 𝜎 ∈Σ ) is freely generated by 𝑋 , for a 𝑋 ⊆ 𝐴 , if for every Σ -hyperalgebra = ( 𝐵, { 𝜎 } 𝜎 ∈Σ ) and map 𝑓 ∶ 𝑋 → 𝐵 there exists only one homomorphism 𝑓 ∶ → extending 𝑓 . In other words, if 𝑗 ∶ 𝑋 → 𝐴 is the inclusion, there exists only one homomorphism 𝑓 ∶ → commutingthe diagram in Set 𝑋 𝑓𝑗 𝑓 Proposition 4.2. If and are freely generated by, respectively, 𝑋 and 𝑌 such that | 𝑋 | = | 𝑌 | , then and are isomorphic.Proof. Since 𝑋 and 𝑌 are of the same cardinality, there exists bijective functions 𝑓 ∶ 𝑋 → 𝑌 and 𝑔 ∶ 𝑌 → 𝑋 inverses of each other. Take the extensions 𝑓 ∶ → and 𝑔 ∶ → and we have that 𝑔 ◦ 𝑓 is an homomorphismextending 𝑔 ◦ 𝑓 = 𝑖𝑑 , the identity on 𝑋 .Since the identical homomorphism 𝐼𝑑 ∶ → also extends 𝑖𝑑 , we have that 𝐼𝑑 = 𝑔 ◦ 𝑓 . In a similar waywe have that 𝐼𝑑 = 𝑓 ◦ 𝑔 , so that and are isomorphic.This way we can refer ourselves to the Σ -hyperalgebra freely generated by 𝑋 , up to isomorphisms.Remember that we have defined MAlg (Σ) as the category whose objects are exactly all Σ -hyperalgebras andfor which, given Σ -hyperalgebras and , 𝐻𝑜𝑚
MAlg (Σ) ( , ) is the set of all homomorphisms from to . Wewill denote by ∶ MAlg (Σ) → Set the forgetful functor.
Lemma 4.3.
The functor 𝐹 ∶ Set → MAlg (Σ) , associating a set 𝑋 to a Σ -hyperalgebra freely generated by 𝑋 ,which we will denote 𝐹 𝑋 , and a function 𝑓 ∶ 𝑋 → 𝑌 to the only homomorphism 𝑓 ∶ 𝐹 𝑋 → 𝐹 𝑌 extending 𝑓 , isa left adjoint of .Proof. For 𝑋 a set and a Σ -hyperalgebra with universe 𝐴 we consider Φ ,𝑋 ∶ 𝐻𝑜𝑚
Set ( 𝑋, ) → 𝐻𝑜𝑚
MAlg (Σ) ( 𝐹 𝑋, ) associating a map 𝑓 ∶ 𝑋 → 𝐴 to the only homomorphism 𝑓 ∶ 𝐹 𝑋 → extending 𝑓 . Each Φ ,𝑋 is clearly abijection given that 𝐹 𝑋 is freely generated by 𝑋 .Now, given sets 𝑋 and 𝑌 , Σ -hyperalgebras and , a function 𝑓 ∶ 𝑌 → 𝑋 and an homomorphism ℎ ∶ → , we have only left to prove that the following diagram commutes in Set . 𝐻𝑜𝑚
Set ( 𝑋, ) 𝐻𝑜𝑚
MAlg (Σ) ( 𝐹 𝑋, ) 𝐻𝑜𝑚
Set ( 𝑌 , ) 𝐻𝑜𝑚
MAlg (Σ) ( 𝐹 𝑌 , ) Φ ,𝑋 𝐻𝑜𝑚 ( 𝑓, 𝜑 ) 𝐻𝑜𝑚 ( 𝐹 𝑓,𝜑 )Φ ,𝑌 So we take a function 𝑔 ∶ 𝑋 → : taking the superior edge of the diagram we have Φ ,𝑋 𝑔 = 𝑔 and 𝐻𝑜𝑚 ( 𝐹 𝑓 , ℎ ) 𝑔 = ℎ ◦ 𝑔 ◦ 𝐹 𝑓 ; on the inferior one,
𝐻𝑜𝑚 ( 𝑓 , ℎ ) 𝑔 = ℎ ◦ 𝑔 ◦ 𝑓 and Φ ,𝑌 ℎ ◦ 𝑔 ◦ 𝑓 = ℎ ◦ 𝑔 ◦ 𝑓 .Now both ℎ ◦ 𝑔 ◦ 𝐹 𝑓 and ℎ ◦ 𝑔 ◦ 𝑓 are homomorphisms from 𝐹 𝑌 to extending ℎ ◦ 𝑔 ◦ 𝑓 ∶ 𝑌 → : forthe second one this is obvious, for the first we take an element 𝑦 ∈ 𝑌 and see that ℎ ◦ 𝑔 ◦ 𝐹 𝑓 ( 𝑦 ) = ℎ ◦ 𝑔 ◦ 𝑓 ( 𝑦 ) = ℎ ◦ 𝑔 ◦ 𝑓 ( 𝑦 ) = ℎ ◦ 𝑔 ◦ 𝑓 ( 𝑦 ) bsolutely Free Hyperalgebras 15since, respectively, 𝐹 𝑓 = 𝑓 , which extends 𝑓 ; 𝑔 extends 𝑔 , which is defined on 𝑋 that contains 𝑓 ( 𝑦 ) ; and ℎ = ℎ ,seen only as a function between sets.Given that 𝐹 𝑌 is freely generated over 𝑌 , we have that ℎ ◦ 𝑔 ◦ 𝐹 𝑓 = 𝜑 ◦ 𝑔 ◦ 𝑓 and the diagram in fact commutes. Theorem 4.4.
Given a non-empty signature Σ and a set 𝑋 , there does not exist a Σ -hyperalgebra freely generatedby 𝑋 .Proof. Suppose that = ( 𝐴, { 𝜎 } 𝜎 ∈Σ ) is freely generated by 𝑋 and let be a set that properly contains 𝑋 ,meaning ≠ ∅ and therefore that T (Σ , ) is well defined; then take the identity function 𝑗 ∶ 𝑋 → 𝑇 (Σ , ) , suchthat 𝑗 ( 𝑥 ) = 𝑥 for every 𝑥 ∈ 𝑋 , and the homomorphism 𝑗 ∶ → T (Σ , ) extending 𝑗 .Now, take the identity function 𝑖𝑑 ∶ → 𝑇 (Σ , ) and the collections of choices 𝐶 and 𝐷 from T (Σ , ) to mT (Σ , , such that, for 𝜎 ∈ Σ 𝑛 , 𝐶𝜎 𝛽 , … ,𝛽 𝑛 𝛼 , … ,𝛼 𝑛 ( 𝜎𝛼 … 𝛼 𝑛 ) = 𝜎 𝛽 … 𝛽 𝑛 and 𝐷𝜎 𝛽 , … ,𝛽 𝑛 𝛼 , … ,𝛼 𝑛 ( 𝜎𝛼 … 𝛼 𝑛 ) = 𝜎 𝛽 … 𝛽 𝑛 , and consider the only homomorphisms 𝑖𝑑 𝐶 , 𝑖𝑑 𝐷 ∶ T (Σ , ) → mT (Σ , , extending, respectively, 𝑖𝑑 and 𝐶 , and 𝑖𝑑 and 𝐷 , which we know to exist given T (Σ , ) is cdf -generated by . Since 𝑖𝑑 𝐶 ◦ 𝑗, 𝑖𝑑 𝐷 ◦ 𝑗 ∶ → mT (Σ , , both extend the function 𝑗 ′ ∶ 𝑋 → 𝑇 (Σ , ) such that 𝑗 ′ ( 𝑥 ) = 𝑥 for every 𝑥 ∈ 𝑋 (remember properly contains 𝑋 ), we have 𝑖𝑑 𝐶 ◦ 𝑗 = 𝑖𝑑 𝐷 ◦ 𝑗 .Now, if 𝛼 ∈ 𝑇 (Σ , ) ⧵ , we have that there exists 𝜎 ∈ Σ 𝑛 , for 𝑛 ∈ ℕ , and elements 𝛼 , … , 𝛼 𝑛 ∈ 𝑇 (Σ , ) suchthat 𝛼 = 𝜎𝛼 … 𝛼 𝑛 ; in this case, 𝑖𝑑 𝐶 ( 𝛼 ) = 𝜎 𝑖𝑑 𝐶 ( 𝛼 ) … 𝑖𝑑 𝐶 ( 𝛼 𝑛 ) ≠ 𝜎 𝑖𝑑 𝐷 ( 𝛼 ) … 𝑖𝑑 𝐷 ( 𝛼 𝑛 ) = 𝑖𝑑 𝐷 ( 𝛼 ) , given that the main connectives are distinct, and therefore implying that 𝑖𝑑 𝐶 and 𝑖𝑑 𝐷 are always different outsideof .Since 𝑖𝑑 𝐶 ◦ 𝑗 = 𝑖𝑑 𝐷 ◦ 𝑗 , we must have that 𝑗 ( 𝐴 ) ⊆ , and this is absurd since we are assuming Σ non-empty: if Σ ≠ ∅ , for a 𝜎 ∈ Σ and 𝑎 ∈ 𝜎 we have that 𝑗 ( 𝑎 ) = 𝜎 in 𝑇 (Σ , ) , which is not in 𝑋 ; if it is another Σ 𝑛 which isnot empty, given 𝑎 ∈ 𝐴 (which exists given the universe of hyperalgebras are assumed to be non-empty) we havethat, for 𝑏 ∈ 𝜎 ( 𝑎, … , 𝑎 ) , is valid that 𝑗 ( 𝑏 ) = 𝜎 ( 𝑗 ( 𝑎 ) , … , 𝑗 ( 𝑎 )) , which is again not in .We must conclude that there is no freely generated hyperalgebras. Corollary 4.5.
The category
MAlg (Σ) does not have an initial object.Proof.
We state that if is an initial object, is freely generated by ∅ : in fact, for every Σ -hyperalgebra and map 𝑓 ∶ ∅ → 𝐵 , there exists a single homomorphism ! ∶ → extending 𝑓 = ∅ , that is, the only homomorphismbetween and .By Theorem 4.4, freely generated hyperalgebras do not exist, what ends the proof. Theorem 4.6.
The forgetful functor ∶ MAlg (Σ) → Set does not have a left adjoint.Proof.
For suppose we have a left adjoint 𝐹 ∶ Set → MAlg (Σ) of , so that 𝐹 has a left adjoint and is thereforecocontinuous. Since ∅ is the initial object in Set , we have that 𝐹 ∅ must be an initial object in MAlg (Σ) , which byCorollary 4.5 does not exist.
Conclusion and Future Work
We believe to have shown here how hyperalgebras of terms have a richness of their own, and deserve to be furtherstudied. Their connections to graph theory and to the theory of partial orders are self-evident, and suggest otherproperties of these objects, and possibly other characterizations. More prominently, they were originally thoughtout in order to facilitate the use of non-deterministic semantics for non-classical logics, specially those of a para-consistent character (see, for instance, [4]). From the present study, we hope to obtain, with the aid of mT (Σ , , 𝜅 ) (now seen as the hyperalgebra of propositional formulas) and its subhyperalgebras, new interpretations of existingsemantics for logic systems and new semantics altogether. Here, of course, decision problems concerning thesehyperalgebras become relevant and need to be addressed.Finally, in what is possibly the most important open question concerning hyperalgebras of terms, we refer backto something we mentioned in various point of this text: in universal algebra, a Σ -algebra is said to be freelygenerated by a subset 𝑋 of its universe in a variety 𝑉 of Σ -algebras when, for every in 𝑉 and every functionbsolutely Free Hyperalgebras 16 𝑓 ∶ 𝑋 → 𝐵 , there exists an unique homomorphism 𝑓 ∶ → extending 𝑓 ; these are the relatively free algebras,which can be obtained in a variety from a quotient of T (Σ , ) . Some questions naturally arise: are there analogousof cdf -generated hyperalgebras with respect to classes, analogous to varieties, of hyperalgebras? If so, are theyobtained in some meaningful way from the hyperalgebras of terms? Acknowledgements
The first author acknowledges support from the National Council for Scientific and Techno-logical Development (CNPq), Brazil under research grant 306530/2019-8. The second author was supported by adoctoral scholarship from CAPES, Brazil. We would also like to thank Hugo Mariano, Darllan Pinto, Peter Arndt,Ana Cláudia Golzio and Kaique Matias for making suggestions that greatly improved the clarity of this exposition.
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