aa r X i v : . [ m a t h . L O ] J un Beth Definability in the Logic KR Jacob GarberJune 15, 2020
Abstract
The Beth Definability Property holds for an algebraizable logic if and only if every epi-morphism in the corresponding category of algebras is surjective. Using this technique,Urquhart in 1999 showed that the Beth Definability Property fails for a wide variety ofrelevant logics, including T , E , and R . However, the counterexample for those logicsdoes not extend to the algebraic counterpart of KR , the Boolean monoids. Followingan approach suggested by Urquhart, we use modular lattices constructed by Freeseto show that epimorphisms need not be surjective in a wide class of relation algebras.This includes the Boolean monoids, which shows that the Beth Definability Propertyfails for KR . Relevant logics were first introduced to avoid the paradoxes of material implication , which arethe result of a mismatch between the intuitive meaning of implication and its formalizationin classical logic. Beginning with the Russian philosopher Ivan Orlov in 1928, this work leadto the development of the logics of ticket entailment T , relevant entailment E , and relevantimplication R . A comprehensive description of these logics and relevant logic in general canbe found in [1, 2, 7, 15, 5]. In this paper we will focus our attention on the logic KR , whichconsists of adding the paradoxical axiom ( A ∧ ¬ A ) → B to R . Despite being stronger than R and thus not a purely relevant logic, KR does not collapse to classical logic. Following Dunnand Restall [7], we call any such logic a super-relevant logic. There are several importanttheorems of classical logic that fail for the relevant logics, one of which we describe below. Definition 1.1.
Let L be a propositional logic and Σ a set of formulas from L containing avariable p . For a new variable q , let Σ[ q/p ] denote the result of replacing all instances of p with q . We say Σ implicitly defines p ifΣ ∪ Σ[ q/p ] ⊢ p ↔ q. Alternatively, we say Σ explicitly defines p if there is a formula A containing only the variablesin Σ without p , such that Σ ⊢ p ↔ A. A logic L is said to have the Beth Definability Property if for any set of formulas Σ andvariable p , if Σ implicitly defines p , then Σ also explicitly defines p .1he well-known Beth Definability Theorem states that the Beth Definability Propertyholds for classical propositional logic. However, as shown by Urquhart [16] in 1999, theBeth Definability Property fails for a wide variety of relevant logics, including T , E , and R . The techniques of that paper rely on the fact that Boolean negation is implicitly (butnot explicitly) definable in those logics, which does not extend to KR . However, Urquhartconjectured that the Beth Definability Property fails for KR as well, and outlined a possiblemethod of attack using algebraic logic.In algebraic logic, every algebraizable logic L has a corresponding category of algebras,denoted Alg L . For instance, the algebras of classical logic are the Boolean algebras, andthose of intuitionistic logic are the Heyting algebras. Using this correspondence, it is oftenpossible to translate properties of a logic into properties of its corresponding algebra. Definition 1.2.
For objects
A, B in a category, we say a map f : A → B is an epimorphism if for any other object C and maps g, h : B → C ,if g ◦ f = h ◦ f, then g = h. The Beth Definability Property holds in an algebraizable logic L iff in the correspondingalgebra Alg L all epimorphisms are surjective (the ES property). This correspondence wasfirst proven by Nmeti in [13, Section 5.6], and further developed by Blok and Hoogland in[4]. Intuitively, one can think of epimorphisms as being implicit definitions, and surjectionsas being explicit ones. For algebras A ⊆ B , we say A is an epic subalgebra of B if theinclusion map i : A → B is an epimorphism. Of course, this inclusion map is surjective iff A = B . To disprove the Beth Definability Property for a logic, it thus suffices to find aproper epic subalgebra in its corresponding category. To apply this to KR , we will analyzeepimorphisms in the category of algebras for KR , the Boolean monoids . Boolean monoids areclosely related to relation algebras, and can be equivalently characterized as dense symmetricrelation algebras.We will tackle this problem using the approach described in Urquhart [17, Problem 5.3].As discussed in that paper, there is a general correspondence between Boolean monoids andmodular lattices. Every Boolean monoid contains a modular lattice, and given a modularlattice L , one can construct a corresponding Boolean monoid A ( L ) that contains an isomor-phic copy of L as a sublattice. As shown by Freese in [8, Theorem 3.3], there exist modularlattices A and B such that A is a proper epic sublattice of B . Using the above correspon-dence, we extend this to the construction of a proper epic sub-Boolean monoid, which showsthe Beth Definability Property fails for KR . This construction is rather general, and in factshows that ES fails for a wide class of relation algebras that includes the Boolean monoids. As shown by Anderson and Belnap in [1, Section 28.2.3], the algebraic counterpart of thelogic R with truth constant t (sometimes denoted R t ) is the variety of De Morgan monoids .These are De Morgan lattices with a commutative monoid operation. The addition of theaxiom ( A ∧ ¬ A ) → B to R corresponds to adding the axiom a ∧ ¬ a = 0 to the algebra, whichcollapses the De Morgan lattice to a Boolean algebra. Such objects, which we call Boolean onoids , are the algebraic counterpart of KR . A particularly useful description of Booleanmonoids is in terms of relation algebras. The following axiomatization of relation algebrasis taken from Givant [11, Definition 2.1]. Definition 2.1. A relation algebra is an algebra (cid:10) A, ∨ , ¬ , ◦ , ` , t (cid:11) such that for all a, b, c ∈ A ,1. a ∨ b = b ∨ a a ∨ ( b ∨ c ) = ( a ∨ b ) ∨ c ¬ ( ¬ a ∨ b ) ∨ ¬ ( ¬ a ∨ ¬ b ) = a a ◦ ( b ◦ c ) = ( a ◦ b ) ◦ c a ◦ t = a a `` = a
7. ( a ◦ b ) ` = b ` ◦ a `
8. ( a ∨ b ) ◦ c = ( a ◦ c ) ∨ ( b ◦ c )9. ( a ∨ b ) ` = a ` ∨ b ` (cid:0) a ` ◦ ¬ ( a ◦ b ) (cid:1) ∨ ¬ b = ¬ b With the standard definition of a ∧ b = ¬ ( ¬ a ∨ ¬ b ), axioms 1–3 imply that h A, ∨ , ∧ , ¬i is a Boolean algebra, axioms 4–7 imply that (cid:10) A, ◦ , ` , t (cid:11) is a monoid with involution, andaxioms 8–10 relate the Boolean and monoid operations to each other. For all a, b ∈ A , arelation algebra A is called1. abelian if a ◦ b = b ◦ a symmetric if a ` = a dense if a ≤ a ◦ a .In particular, a Boolean monoid can be equivalently defined as a dense symmetric relationalgebra .One important result from the theory of relation algebras is that every abelian relationalgebra contains a special set of elements that form a modular lattice. Definition 2.2.
A lattice L is called modular if for all x, y, z ∈ L , x ≤ z = ⇒ x ∨ ( y ∧ z ) = ( x ∨ y ) ∧ z. This implication is equivalent to the following dual identities.( x ∧ y ) ∨ ( x ∧ z ) = x ∧ ( y ∨ ( x ∧ z ))( x ∨ y ) ∧ ( x ∨ z ) = x ∨ ( y ∧ ( x ∨ z ))3 efinition 2.3. For a relation algebra A , an element a ∈ A is called reflexive if t ≤ a , symmetric if a ` = a , and transitive if a ◦ a ≤ a . An element with all three of these propertiesis a reflexive equivalence element. Define L ( A ) to be the set of all reflexive equivalenceelements of A . Theorem 2.4.
The set of reflexive equivalence elements L ( A ) of an abelian relation algebra A is closed under fusion and meet, and forms a bounded modular lattice. Join is given by a ◦ b , meet by a ∧ b , t is the bottom element, and 1 is the top. Proof.
See Givant [11, Corollary 5.17].We can also in some sense reverse the above theorem, and use a modular lattice toconstruct a relation algebra.
Definition 2.5. A KR frame or model structure is a triple F = h S, R, i of a set S with aternary relation R and distinguished element 0, satisfying:1. R ab iff a = b Raaa Rabc implies
Rbac and
Racb (total symmetry)4.
Rabc and
Rcde implies ∃ f ∈ S such that Radf and
Rf be (Pasch’s Postulate)The last condition has close ties to projective geometry and is explored by Urquhart in [17].
Definition 2.6.
For a KR frame F = h S, R, i , the complex algebra of F is the algebra A ( F ) = h P ( S ) , ∪ , ∩ , c , ◦ , t i , where1. h P ( S ) , ∪ , ∩ , c i is the Boolean algebra on the power set of S .2. t = { } is the monoid identity.3. For A, B ⊆ S , fusion is defined as A ◦ B = { c ∈ S | Rabc for some a ∈ A, b ∈ B } . As described in Urquhart [17, Section 2], the complex algebra A ( F ) of a KR frame isa Boolean monoid. We often write A ( S ) for the complex algebra when the ternary relationand distinguished element are understood from context. Definition 2.7.
For a lattice L with least element 0, we define the following ternary relationon the elements of L : Rabc ⇐⇒ a ∨ b = a ∨ c = b ∨ c. Then with this relation, h L, R, i is a KR frame iff L is modular. Proof.
The first three properties follow immediately from the lattice structure of L . The lastproperty is equivalent to the modular law on L , which is shown in Urquhart [17, Theorem2.7]. 4hus for a modular lattice L with 0, the lattice complex algebra A ( L ) is a Booleanmonoid. An alternative more direct proof of this can be found in Givant [11, Section 3.7].The definition of a KR frame is also an instance of the more general notion of a relationalstructure , where the construction of a complex algebra can be repeated in the context ofBoolean algebras with operators. Givant [10, Chapter 19] and [9, Chapter 1] go into moredetail.For a lattice complex algebra A ( L ), a particularly simple description of its reflexiveequivalence elements can be given in terms of the ideals of L . Definition 2.8.
For a lattice L , an ideal of L is a non-empty subset J ⊆ L such that1. If a, b ∈ J , then a ∨ b ∈ J .2. If a ∈ J and b ≤ a , then b ∈ J .The set of all ideals of L is denoted Id L , which forms a lattice with respect to set inclusion.A special class of ideals are the principal ideals , which are of the form ( a ] = { b ∈ L | b ≤ a } .We shall sometimes use the notation ( a ] L to emphasize that this is the principal ideal of a inside L . Proposition 2.9.
For a lattice L , the principal ideal map I : L → Id La ( a ]is an injective homomorphism of lattices. Proof.
See Grtzer [12, Corollary 4, p. 24].
Theorem 2.10.
For a modular lattice L with least element 0, the set of reflexive equivalenceelements L ( A ( L )) and the set of ideals Id L are identical as lattices. That is, L ( A ( L )) = Id L ,and for all ideals J, K ∈ Id L J ∨ K = J ◦ KJ ∧ K = J ∩ K. Proof.
See Maddux [14, p. 243].
The purpose of this section is to prove Theorem 3.2, which states that for all completesublattices K of a modular lattice L , there is a corresponding complete embedding φ : A ( K ) → A ( L ) of Boolean monoids. The construction of this map will rely on the followingresult from the theory of relation algebras. Theorem 3.1.
Let A be a complete and atomic Boolean monoid, U the set of atoms of A , and B a complete Boolean monoid. Suppose φ : U → B is a map with the followingproperties: 5. The elements φ ( u ) for u ∈ U are non-zero, mutually disjoint, and sum to 1 in B .2. t = W { φ ( u ) | u ∈ U and u ≤ t } φ ( u ) ◦ φ ( v ) = W { φ ( w ) | w ∈ U and w ≤ u ◦ v } for all u, v ∈ U .Then φ extends in a unique way to a complete embedding φ : A → B of Boolean monoids,given by φ ( r ) = _ { φ ( u ) | u ∈ U and u ≤ r } , where r is any element of A . Proof.
This is a specialization of Givant [11, Theorem 7.13 and Corollary 7.14] to Booleanmonoids.We apply this to lattice complex algebras as follows. Recall that for a complete lattice L , a subset K ⊆ L is a complete sublattice iff for all S ⊆ K , V S ∈ K and W S ∈ K , wherethese infima and suprema are calculated in L . Theorem 3.2.
Let L be a complete modular lattice, K ⊆ L a complete sublattice, and I K and I L their respective principal ideal maps. Then there exists a unique complete embeddingof Boolean monoids φ : A ( K ) → A ( L ) such that φ ◦ I K = I L . Proof.
We will first show uniqueness to determine what the map φ should be, and then usethat definition to show it is a complete embedding.Suppose that φ : A ( K ) → A ( L ) is a complete embedding with φ ◦ I K = I L . Since φ is acomplete homomorphism, it is determined by its values on the singleton subsets of K , whichare the atoms of A ( K ). For all a ∈ K , we can write ( a ] K as the disjoint union( a ] K = { a } ∪ [ b < ab ∈ K ( b ] K = ⇒ φ (( a ] K ) = φ ( { a } ) ∪ [ b < ab ∈ K φ (( b ] K ) φ is a complete homomorphism= ⇒ ( a ] L = φ ( { a } ) ∪ [ b < ab ∈ K ( b ] L φ ◦ I K = I L = ⇒ φ ( { a } ) = ( a ] L (cid:15) [ b < ab ∈ K ( b ] L φ preserves disjoint unionsThus φ is uniquely determined.So then, let U be the set of singletons in A ( K ), and define the map φ : U → A ( L ) by φ ( { a } ) = ( a ] L (cid:15) [ b < ab ∈ K ( b ] L We will verify the three conditions of Theorem 3.1 to show that this can be extended to acomplete embedding of Boolean monoids. 6. It suffices to show that the sets φ ( { a } ) for a ∈ K are non-empty, mutually disjoint,and cover L . • All sets are non-empty, since a ∈ φ ( { a } ) for any a ∈ K . • Let a, b ∈ K be distinct elements. Then a ∧ b ≤ a , and a ∧ b ≤ b . Since a and b aredistinct, at least one of the previous inequalities must be strict, so without lossof generality suppose a ∧ b < a . Since K is a sublattice, a ∧ b ∈ K . Now suppose x ∈ φ ( { a } ) ∩ φ ( { b } ). Then x ≤ a and x ≤ b , so x ∈ ( a ∧ b ] L . Since a ∧ b < a ,this implies x / ∈ φ ( { a } ), which is a contradiction. Thus φ ( { a } ) and φ ( { b } ) aredisjoint. • For an arbitrary x ∈ L , let a = ^ F x where F x = { b ∈ K | x ≤ b } . Since K is a complete sublattice, this infimumexists and is an element of K . By definition, x is a lower bound for F x , so x ≤ a ,and thus x ∈ ( a ] L . Since a ∈ F x , we in fact have a = min F x . Furthermore, forany other b ∈ K with b < a , it cannot be that x ∈ ( b ] L , since then we would have a ≤ b , which is impossible. Thus x ∈ ( a ] L (cid:15) [ b < ab ∈ K ( b ] L = φ ( { a } ) . So the images of φ cover L .2. The monoid identity t = { } is itself a singleton, and we trivially have φ ( t ) = φ ( { } ) = (0] L = { } = t.
3. From left to right, let a, b ∈ K , and suppose that z ∈ φ ( { a } ) ◦ φ ( { b } ). We wish toshow z ∈ φ ( { c } ), for some c ∈ K with { c } ⊆ { a } ◦ { b } .By assumption Rxyz for some x ∈ φ ( { a } ) and y ∈ φ ( { b } ). From Condition 1, we know a = min F x , b = min F y , and z ∈ φ ( { c } ), where c = min F z . Since x ≤ a and y ≤ b , wehave x ∨ y ≤ a ∨ b = ⇒ x ∨ z ≤ a ∨ b since Rxyz = ⇒ z ≤ a ∨ b = ⇒ c ≤ a ∨ b minimality of c = ⇒ a ∨ c ≤ a ∨ b. Symmetrically, we conclude a ∨ b ≤ a ∨ c , and so a ∨ b = a ∨ c . A similar argumentshows a ∨ c = b ∨ c . Thus Rabc , and so c ∈ { a } ◦ { b } as desired.From right to left, let a, b, c ∈ K and suppose { c } ⊆ { a } ◦ { b } . We wish to show φ ( { c } ) ⊆ φ ( { a } ) ◦ φ ( { b } ). That is, for all z ∈ φ ( { c } ), there exists x ∈ φ ( { a } ) and7 ∈ φ ( { b } ) such that Rxyz . To do this, we use an approach similar to the one ofMaddux in [14, p. 244, part (2)]. For a given z , let x = ( b ∨ z ) ∧ ay = ( a ∨ z ) ∧ b. We first show that a = min F x . From the definition we have a ∧ b ≤ x ≤ a , so a ∈ F x .Now let d ∈ F x be any other element. Then d ∈ K with x ≤ d , so x ≤ a ∧ d = ⇒ x ∨ b ≤ ( a ∧ d ) ∨ b. Furthermore, x ∨ b = (( b ∨ z ) ∧ a ) ∨ b = ( b ∨ z ) ∧ ( a ∨ b ) modularity= ( b ∨ z ) ∧ ( b ∨ c ) since Rabc = b ∨ z since z ≤ c. Since z ≤ b ∨ z = x ∨ b , we then have z ≤ ( a ∧ d ) ∨ b = ⇒ c ≤ ( a ∧ d ) ∨ b minimality of c = ⇒ b ∨ c ≤ ( a ∧ d ) ∨ b = ⇒ a ∨ b ≤ ( a ∧ d ) ∨ b since Rabc.
Using absorption, this implies a ≤ (( a ∧ d ) ∨ b ) ∧ a = ( a ∧ d ) ∨ ( a ∧ b ) modularity= a ∧ d since a ∧ b ≤ x ≤ a ∧ d. Therefore a ≤ d , so a = min F x as wanted. Thus x ∈ φ ( { a } ), and a symmetricargument shows that y ∈ φ ( { b } ).Now we show Rxyz . Using modularity, x ∨ z = (( b ∨ z ) ∧ a ) ∨ z = ( b ∨ z ) ∧ ( a ∨ z )= ( a ∨ z ) ∧ ( b ∨ z )= (( a ∨ z ) ∧ b ) ∨ z = y ∨ z. Since x ≤ a and z ≤ c , we have x ∨ z ≤ a ∨ c = a ∨ b . Thus x ∨ z = ( a ∨ b ) ∧ ( x ∨ z )= ( a ∨ b ) ∧ ( a ∨ z ) ∧ ( b ∨ z ) from above= ( a ∨ ( b ∧ ( a ∨ z ))) ∧ ( b ∨ z ) modularity= ( b ∨ z ) ∧ ( a ∨ (( a ∨ z ) ∧ b )) . a ∨ z ) ∧ b ≤ b ≤ b ∨ z and a final application of the modular law, we thushave x ∨ z = (( b ∨ z ) ∧ a ) ∨ (( a ∨ z ) ∧ b )= x ∨ y. Thus
Rxyz , and the condition is shown.Thus by Theorem 3.1, φ extends uniquely to a complete embedding φ : A ( K ) → A ( L ) ofBoolean monoids, where for all S ⊆ K , φ ( S ) = [ a ∈ S φ ( { a } ) . We use this definition to show that φ ◦ I K = I L . For any a ∈ K , ( a ] K is a reflexive equivalenceelement of A ( K ) by Theorem 2.10. Since φ preserves equational properties, the image φ (( a ] K )is also a reflexive equivalence element of A ( L ), and thus an ideal of L by the same theorem.From the definition of φ , a ∈ φ ( { a } ) ⊆ φ (( a ] K ) , and so ( a ] L ⊆ φ (( a ] K ) from the definition of an ideal.On the other hand, φ (( a ] K ) = [ b ≤ ab ∈ K φ ( { b } ) ⊆ [ b ≤ ab ∈ K ( b ] L = ( a ] L , and so φ (( a ] K ) = ( a ] L . In this section, let
LRA be the class of all subalgebras of lattice complex algebras,
ARA thevariety of abelian relation algebras, and R any class of relation algebras with LRA ⊆ R ⊆ ARA .We will now use modular lattices constructed by Freese and the following general constructionto show that ES fails for R .Let L be a complete modular lattice, and K ⊆ L a complete sublattice. The principalideal map I L : L → Id L is an embedding of lattices, and from Theorem 2.10 we knowId L = L ( A ( L )). Thus, let K ′ = I L ( K ) and L ′ = I L ( L ) be the isomorphic images of L and K contained in L ( A ( L )). In A ( L ), let U be the subalgebra generated by K ′ , and V thesubalgebra generated by L ′ . Theorem 4.1.
In the above situation, if K is a proper epic sublattice of L , then U is aproper R -epic subalgebra of V . Proof.
Since K ′ ⊂ L ′ , U ⊆ V . Let W be any other R -algebra, and f, g : V → W twohomomorphisms that agree on U . The image of a reflexive equivalence element is a reflexiveequivalence element, so f and g restrict to maps f | L ′ , g | L ′ : L ′ → L ( W ) .
9y Theorem 2.4, L ( W ) is a modular lattice under fusion and meet, and f and g preserve theseoperations, so these restrictions are homomorphisms of modular lattices. By assumption, f and g agree on U , and since K ′ ⊆ U they must also agree on K ′ . But K ′ is an epicsublattice of L ′ , so f and g must also agree on L ′ . Thus f | L ′ = g | L ′ , and so f = g since L ′ is the generating set of V . Thus U is an R -epic subalgebra of V .However, U is a proper subalgebra. Let φ : A ( K ) → A ( L ) be the complete embeddingof Theorem 3.2, and let Z = im φ . Then I K ( K ) ⊆ A ( K ), and φ ( I K ( K )) = I L ( K ) = K ′ , so K ′ ⊆ Z , which implies U ⊆ Z since U is the smallest subalgebra that contains K ′ . Nowfor contradiction suppose that U = V . Then L ′ ⊆ V = U ⊆ Z . Then for any x ∈ L , wehave ( x ] L ∈ Z , so there is some S ⊆ K such that φ ( S ) = ( x ] L . In particular then, there isan a ∈ S such that x ∈ φ ( { a } ) ⊆ ( a ] L = ⇒ x ≤ a. On the other hand, a ∈ φ ( { a } ) ⊆ φ ( S ) = ( x ] L = ⇒ a ≤ x. Thus x = a , so x ∈ K . The element x ∈ L was arbitrary, so L = K , which is a contradiction. Theorem 4.2.
ES fails for any class R of relation algebras with LRA ⊆ R ⊆ ARA . Proof.
In [8, Theorem 3.3], Freese constructs modular lattices A ⊂ B such that A is aproper epic sublattice of B . B has no infinite chains, so is complete by Davey and Priestley[6, Theorem 2.41 (iii)]. Likewise, A is a { } -sublattice of B , and as a sublattice is completeby Theorems 2.40 and 2.41 (i) of the same. We can thus apply Theorem 4.1 to A and B ,and the result follows. Corollary 4.3.
ES fails for the varieties of abelian, symmetric, and dense symmetric relationalgebras.
Corollary 4.4.
The Beth Definability Property fails for KR . Using modular lattices constructed by Freese, we have proved that epimorphisms need notbe surjective in a wide class of relation algebras. This includes the Boolean monoids, whichshows that the Beth Definability Property fails for the super-relevant logic KR . This shouldbe contrasted with the result of [3, Theorem 8.5], which shows that the Beth DefinabilityProperty does hold for the super-relevant logic RM . The super-relevant logics thus exhibitmore diversity than the relevant logics, where this property fails uniformly. Acknowledgements
The author would like to thank Katalin Bimb for her never-ending encouragement duringthe process of investigating this problem and writing this paper.10 eferences [1] Alan Ross Anderson and Nuel D. Belnap.
Entailment: The Logic of Relevance andNecessity , Volume 1. Princeton University Press, 1975.[2] Alan Ross Anderson, Nuel D. Belnap, and J. Michael Dunn.
Entailment: The Logic ofRelevance and Necessity , Volume 2. Princeton University Press, 1992.[3] Guram Bezhanishvili, Tommaso Moraschini, and James G. Raftery. Epimorphisms inVarieties of Residuated Structures.
Journal of Algebra , 492:185–211, 2017.[4] W. J. Blok and Eva Hoogland. The Beth Property in Algebraic Logic.
Studia Logica ,83:49–90, 2006.[5] Ross Brady, editor.
Relevant Logics and Their Rivals , Volume 2. Ashgate, 2003.[6] B. A. Davey and H. A. Priestley.
Introduction to Lattices and Order . CambridgeUniversity Press, Second Edition, 2002.[7] J. Michael Dunn and Gregory Restall.
Relevance Logic , Volume 6. Springer Netherlands,Second edition, 2002.[8] Ralph Freese. The Variety of Modular Lattices is Not Generated by its Finite Members.
Transactions of the American Mathematical Society , 255:277–300, 1979.[9] Steven Givant.
Duality Theories for Boolean Algebras with Operators . Springer Mono-graphs in Mathematics. Springer International Publishing, 2014.[10] Steven Givant.
Advanced Topics in Relation Algebras . Springer International Publishing,2017.[11] Steven Givant.
Introduction to Relation Algebras . Springer International Publishing,2017.[12] George Grtzer.
General Lattice Theory . Birkuser Basel, Second Edition, 2003.[13] Leon Henkin, J. Donald Monk, and Alfred Tarski.
Cylindric Algebras , Volume 2. North-Holland, 1985.[14] Roger Maddux. Embedding Modular Lattices into Relation Algebras.
Algebra Univer-salis , 12:242–246, 1981.[15] Richard Routley, Robert K. Meyer, Val Plumwood, and Ross T. Brady.
Relevant Logicsand Their Rivals , Volume 1. Ridgeview Publishing Company, 1982.[16] Alasdair Urquhart.
Beth’s Definability Theorem in Relevant Logics , Volume 24 of
Studiesin Fuzziness and Soft Computing . Physica-Verlag Heidelberg, 1999.[17] Alasdair Urquhart. The Geometry of Relevant Implication.