aa r X i v : . [ m a t h . R A ] J a n Sheffer operation in relational systems
Ivan Chajda and Helmut L¨anger
Abstract
The concept of a Sheffer operation known for Boolean algebras and orthomod-ular lattices is extended to arbitrary directed relational systems with involution.It is proved that to every such relational system there can be assigned a Sheffergroupoid and also, conversely, every Sheffer groupoid induces a directed relationalsystem with involution. Hence, investigations of these relational systems can betransformed to the treaty of special groupoids which form a variety of algebras.If the Sheffer operation is also commutative then the induced binary relation isantisymmetric. Moreover, commutative Sheffer groupoids form a congruence dis-tributive variety. We characterize symmertry, antisymmetry and treansitivity ofbinary relations by identities and quasi-identities satisfied by an assigned Shefferoperation. The concepts of twist-products of relational systems and of Kleene re-lational systems are introduced. We prove that every directed relational systemcan be embedded into a directed relational system with involution via the twist-product construction. If the relation in question is even transitive, then the directedrelational system can be embedded into a Kleene relational system. Any Sheffer op-eration assigned to a directed relational system A with involution induces a Shefferoperation assigned to the twist-product of A . AMS Subject Classification:
Keywords:
Relational system, directed relational system, involution, Sheffer operation,Sheffer groupoid, twist-product, Kleene relational system
Relational systems form one of the most general mathematical structures. Almost allstructures appearing in algebra can be considered as relational structures. Such structureswere studied for a long lime, see the pioneering work by J. Riguet ([13]) from 1948containing elementary properties and constructions with binary relations and the paperby R. Fraiss´e ([11]) from 1954. On the other hand, in contrast to publications in algebra,not so many of papers are devoted to relational systems. One of the reasons is thatthere are not so powerful tools for investigating relations as there are for algebras. Thisis also the reason why relational systems do not appear so often in applications both Support of the research by the Austrian Science Fund (FWF), project I 4579-N, and the CzechScience Foundation (GA ˇCR), project 20-09869L, entitled “The many facets of orthomodularity”, as wellas by ¨OAD, project CZ 02/2019, entitled “Function algebras and ordered structures related to logic anddata fusion”, and, concerning the first author, by IGA, project PˇrF 2020 014, is gratefully acknowledged.
1n mathematics and outside. One important application of relational systems are e.g.Kripke systems used in the formalization of several non-classical logical systems.The authors introduced formerly several methods where relational systems are connectedwith various accompanying algebras and hence their properties can be transformed intoalgebraic language and the problems are solved by tools developed in general algebra.Let us mention e.g. [6] and [7] where certain groupoids similar to directoids are assignedor [8] and [10] where this approach is applied to relational systems equipped with aunary operation. For ternary relations, such an approach was used in [5]. However, thespectrum of used algebraic tools is not restricted only to these more or less elementarycases, in [2] relational systems are treated similarly as residuated ordered sets. In thepresent paper we extend this list of used tools by the so-called Sheffer operation.Remember that the Sheffer operation introduced by H. M. Sheffer ([14]) in 1913 was usedin Boolean algebras as a very successful tool since this operation can replace all otherBoolean operations. Namely, every Boolean operation, both basic or derived, can beexpressed by repeatedly using the Sheffer operation, see e.g. [1]. In today terminology,the clone of Boolean functions is generated by the Sheffer operation. This has a surprisingand very successful application in technology because in switching circles, in particular incomputer processors, it suffices to use only one binary operation, namely the Sheffer one.Then the technology of production of such chips is much easier and cheaper than it was inthe beginning of computer era when several parts of the computer were composed by atleast two different kinds of diodes (e.g. one for conjunction and the other one for negation).As it was shown by the first author in [3], a Sheffer operation can be introduced not onlyin Boolean algebras but also in orthomodular lattices or even in ortholattices (see [1] forthese concepts). These algebras form an algebraic axiomatization of the logic of quantummechanics. Since not all authors agreed that such lattices are suitable for modeling thepropositional calculus of the logic of quantum mechanics, they recognized that disjunctionin this logic not necessarily exists for all elements, i.e. that the supremum of two elementsneed not exist if the these elements are not orthogonal. Hence so-called orthomodularposets and orthoposets were introduced. This was the reason why the concept of Shefferoperation was transferred from ortholattices to orthomodular posets and orthoposets, or,more generally, to ordered sets with an involution or a complementation, see [4].The next natural step is to extend this method from posets to more general relationalsystems. In order to avoid difficulties with not everywhere defined operations and someother drawbacks, we consider so-called directed relational systems where the relation isreflexive and equipped with a unary involution operation. The authors show that also inthis case a kind of Sheffer operation can be introduced and the corresponding groupoidcharacterizes the given relational system. The benefit of this method is two-fold. Atfirst, we show that similarly as for Boolean algebras, using an assigned Sheffer operationwe can conversely recover not only the involution but also the given binary relation.Hence the Sheffer operation reduces the type of the relational system and substitutes abinary relation by an everywhere defined operation. And secondly, we show that somebasic properties of binary relations can be characterized with advantage by using thisoperation and that the Sheffer operation enables also more advanced constructions forrelational systems not considered so far. 2
Basic concepts
The Sheffer operation was introduced by H. M. Sheffer ([14]) in Boolean algebras. If B = ( B, ∨ , ∧ , ′ , ,
1) is a Boolean algebra and one defines x | y := x ′ ∨ y ′ then | is just the Sheffer operation on B . For our reasons, we define it as follows. Definition 2.1. A Sheffer operation on a non-void set A is a binary operation | on A satisfying the following identities: ( x | y ) | ( x | x ) ≈ x, (1)( x | y ) | ( y | y ) ≈ y. (2) A Sheffer groupoid is a groupoid ( A, | ) where | is a Sheffer operation on A . Hence, the class of Sheffer groupoids forms a variety of algebras.
Example 2.2. If A := { a, b, c, d } and the binary operation | on A is defined by | a b c da a c d cb c b d cc a b d cd a b d c then ( A, | ) is a Sheffer groupoid. It is worth noticing that the Sheffer operation in a Boolean algebra satisfies the identities(1) and (2) and hence our new concept is sound.An antitone involution on a lattice ( L, ∨ , ∧ ) is a unary operation ′ on L satisfying(i) x ′′ ≈ x ,(ii) x ≤ y implies y ′ ≤ x ′ for all x, y ∈ L .The following lemma was shown for ortholattices in [3]. Lemma 2.3.
Let ( L, ∨ , ∧ , ′ ) be a lattice with an antitone involution. Then (i) and (ii) hold: (i) If x | y := x ′ ∨ y ′ for all x, y ∈ L then ( L, | ) is a Sheffer groupoid. (ii) If x | y := x ′ ∧ y ′ for all x, y ∈ L then ( L, | ) is a Sheffer groupoid.Proof. x | x ≈ x ′ ∨ x ′ ≈ x ′ , (1) and (2) are equivalent to( x ′ ∨ y ′ ) ′ ∨ x ′′ ≈ x, ( x ′ ∨ y ′ ) ′ ∨ y ′′ ≈ y, respectively.(ii) Since x | x ≈ x ′ ∧ x ′ ≈ x ′ , (1) and (2) are equivalent to( x ′ ∧ y ′ ) ′ ∧ x ′′ ≈ x, ( x ′ ∧ y ′ ) ′ ∧ y ′′ ≈ y, respectively. Lemma 2.4.
Axioms (1) and (2) are independent.Proof. if A := { a, b } and the binary operation | on A is defined by x | y := x for all x ∈ A then | satisfies (1), but not (2) since ( a | b ) | ( b | b ) = a | b = a = b , and if A := { a, b, c } andthe binary operation | on A is defined by | a b ca a b cb c b cc a a c then | satisfies (2), but not (1) since ( a | b ) | ( a | a ) = b | a = c = a .Let us recall some concepts from theory of relations.Let A be a non-void set, a, b ∈ A , R a binary relation on A and ′ a unary operation on A . We define U ( a, b ) := { x ∈ A | ( a, x ) , ( b, x ) ∈ R } ,L ( a, b ) := { x ∈ A | ( x, a ) , ( x, b ) ∈ R } and call these set the upper cone and lower cone of a and b with respect to R , respectively.The relational system A = ( A, R ) is called directed if U ( x, y ) = ∅ and L ( x, y ) = ∅ forall x, y ∈ A . The operation ′ is called antitone if ( x, y ) ∈ R implies ( y ′ , x ′ ) ∈ R and an involution on A if it is antitone and if it satisfies the identity x ′′ ≈ x . It can be shownthat in a relational system ( A, R, ′ ) with involution, if U ( x, y ) = ∅ for all x, y ∈ A then L ( x, y ) = ∅ for all x, y ∈ A since L ( x, y ) ≈ ( U ( x ′ , y ′ )) ′ , where B ′ := { b ′ | b ∈ B } for everysubset B of A . Definition 2.5. A directed relational system with involution is an ordered triple ( A, R, ′ ) consisting of a non-void set A , a binary relation R on A and a unary operation ′ on A satisfying the following conditions: R is reflexive , (3)( A, R ) is directed , (4) ′ is an involution on ( A, R ) . (5)4 Representation of relational systems by Sheffergroupoids
The following result shows how a Sheffer groupoid is connected with a directed relationalsystem with involution.
Theorem 3.1.
Let A = ( A, | ) be a Sheffer groupoid and define a unary operation ′ on A and a binary relation R on A by x ′ := x | x for all x ∈ A,R := { ( x, y ) ∈ A | x ′ | y ′ = y } . Then R ( A ) := ( A, R, ′ ) is a directed relational system with involution, the so-called di-rected relational system with involution induced by A .Proof. Let a, b ∈ A . (1) implies x ′′ ≈ x and that R is reflexive. (1) and (2) can bewritten in the equivalent form ( x | y ) | x ′ ≈ x and ( x | y ) | y ′ ≈ y , respectively. If ( a, b ) ∈ R then a ′ | b ′ = b and hence b | a = ( a ′ | b ′ ) | a = a ′ , i.e. ( b ′ , a ′ ) ∈ R showing that ′ is an involutionon ( A, R ). Since ( a ′ | b ′ ) | a = a ′ and ( a ′ | b ′ ) | b = b ′ we have (( a ′ | b ′ ) ′ , a ′ ) , (( a ′ | b ′ ) ′ , b ′ ) ∈ R andhence ( a, a ′ | b ′ ) , ( b, a ′ | b ′ ) ∈ R , i.e. a ′ | b ′ ∈ U ( a, b ) which shows U ( a, b ) = ∅ proving that( A, R ) is directed.
Example 3.2. ( A, A \ { ( a, b ) , ( b, a ) } , ′ ) where a ′ = a , b ′ = b , c ′ = d and d ′ = c is thedirected relational system induced by the Sheffer groupoid A from Example 2.2. In the following we show that also conversely, to every directed relational system withinvolution a Sheffer groupoid can be assigned.Let A = ( A, R, ′ ) be a directed relational system with involution. Define a binary opera-tion | on A as follows: Put x | y := y ′ if ( x ′ , y ′ ) ∈ R and let x | y be an arbitrary element of U ( x ′ , y ′ ) otherwise ( x, y ∈ A ). Then | will be called an operation assigned to A . Lemma 3.3.
Let A = ( A, R, ′ ) be a directed relational system with involution and | abinary operation on A . Then | is assigned to A if and only if (i) ( x, y ) ∈ R if and only if x ′ | y ′ = y , (ii) x | y ∈ U ( x ′ , y ′ ) for all x, y ∈ A .Proof. Let a, b ∈ A . First assume | to be assigned to A . If ( a, b ) ∈ R then ( a ′′ , b ′′ ) ∈ R and hence a ′ | b ′ = b ′′ = b . Conversely, assume a ′ | b ′ = b . Then ( a, b ) / ∈ R would imply ( a ′′ , b ′′ ) / ∈ R andhence b = a ′ | b ′ ∈ U ( a ′′ , b ′′ ) = U ( a, b ) and hence ( a, b ) ∈ R , a contradiction. Hence ( a, b ) ∈ R . This shows (i). If ( a ′ , b ′ ) ∈ R then a | b = b ′ ∈ U ( a ′ , b ′ ). Otherwise, a | b ∈ U ( a ′ , b ′ ), too.This shows (ii). Conversely, if | satisfies (i) and (ii) then clearly | is assigned to A .It should be remarked that if ( A, R, ′ ) is a directed relational system with involution and | an assigned operation then condition (ii) of Lemma 3.3 is equivalent to( x | y ) | ( x | y ) ∈ L ( x, y ) for all x, y ∈ A. In the following we will often use this lemma. Now we prove the converse of Theorem 3.1.5 heorem 3.4.
Let A = ( A, R, ′ ) be a directed relational system with involution and | an operation assigned to A . Then | is a Sheffer operation, a so-called Sheffer operationassigned to A , i.e. G ( A ) := ( A, | ) is a Sheffer groupoid, a so-called Sheffer groupoidassigned to A .Proof. Let a, b ∈ A . Since ( x ′ , x ′ ) ∈ R we have x | x ≈ x ′ . If ( a ′ , b ′ ) ∈ R then ( b, a ) ∈ R and hence ( a | b ) | a ′ = b ′ | a ′ = a and ( a | b ) | b ′ = b ′ | b ′ = b . If ( a ′ , b ′ ) / ∈ R then a | b ∈ U ( a ′ , b ′ )and hence ( a ′ , a | b ) , ( b ′ , a | b ) ∈ R which implies (( a | b ) ′ , a ) , (( a | b ) ′ , b ) ∈ R , i.e. ( a | b ) a ′ = a and ( a | b ) b ′ = b . Remark 3.5.
In general, G ( A ) is not uniquely defined. However, it contains all theinformation on the directed relational system A with involution. In other words, the givendirected relational system with involution can be completely recovered from an assignedSheffer groupoid, see the following result. Theorem 3.6.
Let A = ( A, R, ′ ) be a directed relational system with involution. Then R ( G ( A )) = A .Proof. If G ( A ) = ( A, | ) , R ( G ( A )) = ( A, S, ∗ )then according to the proof of Lemma 3.3, S = { ( x, y ) ∈ A | x ′ | y ′ = y } = { ( x, y ) ∈ A | ( x, y ) ∈ R } = R,x ∗ ≈ x | x ≈ x ′ . On the other hand, we can show for which pairs of elements a Sheffer operation assignedto R ( A, | ) coincides with the Sheffer operation | of a given Sheffer groupoid ( A, | ). Theorem 3.7.
Let A = ( A, | ) be a Sheffer groupoid and G ( R ( A )) = ( A, ◦ ) . Then x ◦ y = x | y if x | y = y | y .Proof. If R ( A ) = ( A, R, ′ ) then any of the following assertions implies the next one: x | y = y | y,x | y = y ′ , ( x ′ , y ′ ) ∈ R,x ◦ y = y ′ ,x ◦ y = x | y. In fact, ◦ need not coincide with | as can be seen by the following example.6 xample 3.8. If | is the Sheffer operation from Example 2.2 then ◦ has the operationtable ◦ a b c da a x d cb y b d cc a b d cd a b d c where x, y ∈ { c, d } since U ( a, b ) = { c, d } in the induced relational system. Hence, if wetake x = d or y = d then ◦ differs from | . We have shown that directed relational systems with involution are nearly in a one-to-onecorrespondence with Sheffer groupoids. Analogously as for Boolean algebras where theSheffer operation substitutes all other operations since they can be derived from it, alsohere the Sheffer operation substitutes both the binary relation and the unary operation.Hence it enables us to reduce the type of the directed relational system with involution.
In the following we characterize some of properties of the relation R of a directed relationalsystem A = ( A, R, ′ ) with involution by means of identities and quasi-identities for aSheffer operation assigned to A . Theorem 4.1.
Let A = ( A, R, ′ ) be a directed relational system with involution and | anassigned Sheffer operation. Then R is symmetric if and only if | satisfies the identity (( x | y ) | ( x | y )) | x ≈ x | x. (6) Proof. If R is symmetric then any of the following assertions implies the next one: x | y ∈ U ( x ′ , y ′ ) , ( x ′ , x | y ) ∈ R, ( x | y, x ′ ) ∈ R, ( x | y ) ′ | x ≈ x ′ , (( x | y ) | ( x | y )) | x ≈ x | x. If, conversely, | satisfies identity (6) then any of the following assertions implies the nextone: ( x, y ) ∈ R,x ′ | y ′ = y,y ′ | x ′ = ( x ′ | y ′ ) ′ | x ′ = x, ( y, x ) ∈ R. Another important property of a binary relation is antisymmetry. Recall that a binaryrelation R is antisymmetric if ( x, y ) , ( y, x ) ∈ R implies x = y .7 heorem 4.2. Let A = ( A, R, ′ ) be a directed relational system with involution and | aSheffer operation assigned to it. Then the following hold: (i) R is antisymmetric if and only if x | y = y ′ and y | x = x ′ imply x = y . (ii) If x | y ≈ y | x then R is antisymmetric.Proof. (i) is clear.(ii) This follows from (i) since x | y ≈ y | x , x | y = y ′ and y | x = x ′ imply x = ( y | x ) ′ =( x | y ) ′ = y .Transitivity of a binary relation can be expressed by an identity for an assigned Shefferoperation as follows. Theorem 4.3.
Let A = ( A, R, ′ ) be a directed relational system with involution and | anassigned Sheffer operation. Then R is transitive if and only if | satisfies the identity x | ((( x | y ) | ( x | y )) | z ) | ((( x | y ) | ( x | y )) | z ) ≈ (( x | y ) | ( x | y )) | z. (7) Proof. If R is transitive then any of the following assertions implies the next one: x | y ∈ U ( x ′ , y ′ ) and ( x | y ) ′ | z ∈ U ( x | y, z ′ ) , ( x ′ , x | y ) , ( x | y, ( x | y ) ′ | z ) ∈ R, ( x ′ , ( x | y ) ′ | z ) ∈ R,x | (( x | y ) ′ | z ) ′ ≈ ( x | y ) ′ | z,x | ((( x | y ) | ( x | y )) | z ) | ((( x | y ) | ( x | y )) | z ) ≈ (( x | y ) | ( x | y )) | z. If, conversely, | satisfies identity (7) then any of the following assertions implies the nextone: ( x, y ) , ( y, z ) ∈ R,x ′ | y ′ = y and y ′ | z ′ = z,x ′ | z ′ = x ′ | ( y ′ | z ′ ) ′ = x ′ | (( x ′ | y ′ ) ′ | z ′ ) ′ = ( x ′ | y ′ ) ′ | z ′ = y ′ | z ′ = z, ( x, z ) ∈ R. Let us introduce the following concepts. A bounded relational system with involution is anordered quintuple A = ( A, R, ′ , ,
1) such that (
A, R, ′ ) is a directed relational system withinvolution, 0 , ∈ A and (0 , x ) , ( x, ∈ R hold for all x ∈ A . A is called complemented if itis bounded and if U ( x, x ′ ) ≈ ≈ ′ . In such a case L ( x, x ′ ) ≈
0. Also these properties ofrelational systems can be characterized by identities and quasi-identities for an assignedSheffer operation. 8 heorem 4.4.
Let ( A, R, ′ ) be a directed relational system with involution and | a Shefferoperation assigned to it. Moreover, let , ∈ A and put A := ( A, R, ′ , , . Then thefollowing hold: (i) A is bounded if and only if it satisfies the identities (0 | | x ≈ x | x and x | (1 | ≈ . (ii) A is complemented if it is bounded, | ≈ and if for every x, y ∈ A , x | ( y | y ) = ( x | x ) | ( y | y ) = y implies y = 1 . Proof. (i) The assertions (0 , x ′ ) ∈ R and ( x ′ , ∈ R are equivalent to 0 ′ | x ≈ x ′ and x | ′ ≈ y ∈ U ( x, x ′ ) , ( x, y ) , ( x ′ , y ) ∈ R,x | y ′ = x ′ | y ′ = y,x | ( y | y ) = ( x | x ) | ( y | y ) = y. As mentioned in Section 2, the class of Sheffer groupoids forms a variety V . We can askone more condition, namely commutativity of | . As shown in Theorems 3.6 and 4.2, thedirected relational systems with involution induced by commutative Sheffer groupoidswill have antisymmetric binary relations. We present a subvariety of V containing allcommutative Sheffer groupoids which has an important congruence property.We recall that a variety V of algebras is called congruence distributive if every memberof V has a distributive congruence lattice. Theorem 4.5.
The variety of Sheffer groupoids ( A, | ) satisfying the identities ( x | y ) | ( x | x ) ≈ ( x | x ) | ( x | y ) , (8)( x | y ) | ( y | y ) ≈ ( y | y ) | ( x | y ) (9) is congruence distributive.Proof. If x ′ := x | x and m ( x, y, z ) := (( x | y ) | ( x | z )) ′ | ( y | z ) then m ( x, z, z ) ≈ (( x | z ) | ( x | z )) ′ | ( z | z ) ≈ ( x | z ) | z ′ ≈ z by (2) ,m ( x, y, x ) ≈ (( x | y ) | ( x | x )) ′ | ( y | x ) ≈ x ′ | ( y | x ) ≈ x by (1), (9) and (2) ,m ( x, x, z ) ≈ (( x | x ) | ( x | z )) ′ | ( x | z ) ≈ x ′ | ( x | z ) ≈ x by (8) and (1) . Kleene relational systems and twist-products
At first, we show how homomorphisms of Sheffer groupoids are related with homomor-phisms of induced directed relational systems with involution. Because in the literaturethere are different concepts of homomorphism of relational systems, we recall the followingone.Let (
A, R ) and (
B, S ) be relational systems. A mapping f : A → B is called a homomor-phism from ( A, R ) to (
B, S ) if( x, y ) ∈ R implies ( f ( x ) , f ( y )) ∈ S. A homomorphism f is called strong if( x, y ) ∈ R if and only if ( f ( x ) , f ( y )) ∈ S. If (
A, R, ′ ) and ( B, S, ∗ ) are relational systems with unary operation then f is a homomor-phism from ( A, R, ′ ) to ( B, S, ∗ ) if it is a homomorphism from ( A, R ) to (
B, S ) satisfying f ( x ′ ) = ( f ( x )) ∗ for all x ∈ A. Theorem 5.1.
Let A = ( A, | A ) and B = ( B, | B ) be Sheffer groupoids and f a homomor-phism from A to B . Then f is a homomorphism between the induced directed relationalsystems R ( A ) and R ( B ) with involution.Proof. Let a, b ∈ A , R ( A ) = ( A, R, ′ ) and R ( B ) = ( B, S, ∗ ). We have f ( x ′ ) ≈ f ( x | A x ) ≈ f ( x ) | B f ( x ) ≈ ( f ( x )) ∗ and hence any of the following assertions implies the next one:( a, b ) ∈ R,a ′ | A b ′ = b,f ( a ′ | A b ′ ) = f ( b ) ,f ( a ′ ) | B f ( b ′ ) = f ( b ) , ( f ( a )) ∗ | B ( f ( b )) ∗ = f ( b ) , ( f ( a ) , f ( b )) ∈ S. For the converse direction, we firstly mention the following result for bounded relationalsystems.
Lemma 5.2.
Let ( A, R, ′ , A , A ) and ( B, S, ∗ , B , B ) be bounded relational systems withinvolution and f a strong homomorphism from A = ( A, R, ′ ) to B = ( B, S, ∗ ) . Furtherassume that f (1 A ) = 1 B . Define binary operations | A and | B on A and B , respectively,by x | A y := (cid:26) y ′ if ( x ′ , y ′ ) ∈ R, A otherwise x | B y := (cid:26) y ∗ if ( x ∗ , y ∗ ) ∈ S, B otherwiseThen ( A, | A ) and ( B, | B ) are Sheffer groupoids assigned to A and B , respectively, and f is a homomorphism from ( A, | A ) to ( B, | B ) . roof. Let a, b ∈ A . Obviously, ( A, | A ) and ( B, | B ) are Sheffer groupoids. If ( a ′ , b ′ ) ∈ R then (( f ( a )) ∗ , ( f ( b )) ∗ ) = ( f ( a ′ ) , f ( b ′ )) ∈ S and hence f ( a | A b ) = f ( b ′ ) = ( f ( b )) ∗ = f ( a ) | B f ( b ). If ( a ′ , b ′ ) / ∈ R then (( f ( a )) ∗ , ( f ( b )) ∗ ) = ( f ( a ′ ) , f ( b ′ )) / ∈ S and hence f ( a | A b ) = f (1 A ) = 1 B = f ( a ) | B f ( b ).We are going to determine conditions under which the converse of Theorem 5.1 holds. Theorem 5.3.
Let A = ( A, R, ′ ) and B = ( B, S, ∗ ) be directed relational systems withinvolution, f a strong surjective homomorphism from A to B and | A a Sheffer operationassigned to A and assume that the equivalence relation ker f on A is a congruence on ( A, | A ) . Then there exists a Sheffer operation | B on B such that f is a homomorphismfrom ( A, | A ) to ( B, | B ) and | B is assigned to B .Proof. Define f ( x ) | B f ( y ) := f ( x | A y ) for all x, y ∈ A . Since ker f ∈ Con( A, | A ), | B iswell-defined. Let a, b ∈ A . Then any of the following assertions implies the next one:(( f ( a )) ∗ , ( f ( b )) ∗ ) ∈ S, ( f ( a ′ ) , f ( b ′ )) ∈ S, ( a ′ , b ′ ) ∈ R,a | A b = b ′ ,f ( a | A b ) = f ( b ′ ) ,f ( a ) | B f ( b ) = ( f ( b )) ∗ . Moreover, any of the following assertions implies the next one:(( f ( a )) ∗ , ( f ( b )) ∗ ) / ∈ S, ( f ( a ′ ) , f ( b ′ )) / ∈ S, ( a ′ , b ′ ) / ∈ R,a | A b ∈ U ( a ′ , b ′ ) ,f ( a | A b ) ∈ U ( f ( a ′ ) , f ( b ′ )) ,f ( a ) | B f ( b ) ∈ U (( f ( a )) ∗ , ( f ( b )) ∗ ) . This shows that | B is a Sheffer operation on B assigned to B . According to the definitionof | B we have f ( x | A y ) = f ( x ) | B f ( y ) for all x, y ∈ A .For a lattice L = ( L, ∨ , ∧ ) its twist-product ( L , ⊔ , ⊓ ) is defined by( x, y ) ⊔ ( z, v ) := ( x ∨ z, v ∧ y ) , ( x, y ) ⊓ ( z, v ) := ( x ∧ z, v ∨ y )for all ( x, y ) , ( z, v ) ∈ L . We extend this concept to relational systems as follows.Let A be a non-void set and R a binary relation on A . Then ( A , S, ∗ ) with S := { (( x, y ) , ( z, v )) ∈ ( A ) | ( x, z ) , ( v, y ) ∈ R } , ( x, y ) ∗ := ( y, x )for all ( x, y ) ∈ A will be called the twist-product of ( A, R ).Recall that an embedding of a relational system A into a relational system B is an injectivestrong homomorphism from A to B .The importance of twist-products is illuminated by the next result.11 heorem 5.4. Let A = ( A, R ) be a relational system, a ∈ A and B = ( A , S, ∗ ) thetwist-product of A . Then the following hold: (i) If A is directed then B is a directed relational system with involution ∗ , (ii) the mapping x ( x, a ) is an embedding of A into ( A , S ) .Proof. Let a, b, c, d ∈ A .(i) Assume A to be directed. Since ( a, a ) , ( b, b ) ∈ R we have (( a, b ) , ( a, b )) ∈ S showingreflexivity of S . Because of U (( a, b ) , ( c, d )) = U ( a, c ) × L ( b, d ), ( A , S ) is directed.Moreover, ( x, y ) ∗∗ ≈ ( y, x ) ∗ ≈ ( x, y ), and the following are equivalent:(( a, b ) , ( c, d )) ∈ S, ( a, c ) , ( d, b ) ∈ R, ( d, b ) , ( a, c ) ∈ R, (( d, c ) , ( b, a )) ∈ S, (( c, d ) ∗ , ( a, b ) ∗ ) ∈ S. Hence, ∗ is an involution on ( A , S ).(ii) The mapping x ( x, a ) is injective. Moreover, (( b, a ) , ( c, a )) ∈ S if and only if( b, c ) ∈ R .Hence, every directed relational system can be embedded into a directed relational systemwith involution.The question arises whether a Sheffer operation assigned to the twist-product of a directedrelational A with involution can be derived from a Sheffer operation assigned to A . Wegive a positive answer in the following theorem. Theorem 5.5.
Let ( A, R, ′ ) be a directed relational system with involution, | A an assignedSheffer operation on A and define ( x, y ) | B ( z, v ) := ( y ′ | A v ′ , ( x | A z ) ′ ) for all ( x, y ) , ( z, v ) ∈ A . Then | B is a Sheffer operation on A assigned to the twist-product of ( A, R ) .Proof. For a, b, c, d ∈ A the following are equivalent:(( a, b ) ∗ , ( c, d ) ∗ ) ∈ S, (( b, a ) , ( d, c )) ∈ S, ( b, d ) , ( c, a ) ∈ R, ( b ′′ , d ′′ ) , ( a ′ , c ′ ) ∈ R, ( b ′ | A d ′ , a | A c ) = ( d ′′ , c ′ ) , ( b ′ | A d ′ , ( a | A c ) ′ ) = ( d, c ) , ( a, b ) | B ( c, d ) = ( d, c ) , ( a, b ) | B ( c, d ) = ( c, d ) ∗ b ′ | A d ′ , a | A c ) ∈ U ( b ′′ , d ′′ ) × U ( a ′ , c ′ ) , ( b ′ | A d ′ , ( a | A c ) ′ ) ∈ U ( b, d ) × L ( a, c ) , ( b ′ | A d ′ , ( a | A c ) ′ ) ∈ U (( b, a ) , ( d, c )) , ( a, b ) | B ( c, d ) ∈ U (( a, b ) ∗ , ( c, d ) ∗ ) . In order to simplify notation we extend binary relations between elements of a non-voidset A to relations between subsets of A .Let A be a non-void set, b, c be elements of A , B, C be subsets of A and R be a binaryrelation on A . We say ( B, C ) ∈ R if B × C ⊆ R . Instead of ( { b } , C ) ∈ R and ( B, { c } ) ∈ R we shortly write ( b, C ) ∈ R and ( B, c ) ∈ R , respectively.The concept of a Kleene lattice was introduced by J. A. Kalman ([12]). Recall thata distributive lattice ( L, ∨ , ∧ , ′ ) with antitone involution is called a Kleene lattice if itsatisfies the so-called normality condition , i.e. the identity x ∧ x ′ ≤ y ∨ y ′ for all x, y ∈ L. These lattices are used in logic in order to formalize certain De Morgan propositionallogics. For posets with involution, this notion was already generalized by the authors in[9] in the following way: A distributive poset ( P, ≤ , ′ ) with involution is called a Kleeneposet if L ( x, x ′ ) ≤ U ( y, y ′ ) for all x, y ∈ P which means that z ≤ v for all x, y ∈ P and all ( z, v ) ∈ L ( x, x ′ ) × U ( y, y ′ ). Definition 5.6. (i) A Kleene relational system is a relational system ( A, R, ′ ) with an antitone involu-tion satisfying ( L ( x, x ′ ) , U ( y, y ′ )) ∈ R for all x, y ∈ A. (ii) If A = ( A, R ) is a relational system, a ∈ A and ( A , S, ∗ ) the twist-product of A then we define the following subset of A : P a ( A ) := { ( x, y ) ∈ A | ( L ( x, y ) , a ) , ( a, U ( x, y )) ∈ R } . It is worth noticing that Kleene lattices and Kleene posets are Kleene relational systemsaccording to our previous definition.Using the above defined subset of the twist-product, we can show that every directedrelational system with a transitive relation can be embedded into a Kleene relationalsystem.
Theorem 5.7.
Let A = ( A, R ) be a directed relational system, a ∈ A , ( A , S, ∗ ) thetwist-product of A and T := S ∩ ( P a ( A )) . Then the following hold: (i) If R is transitive then ( P a ( A ) , T, ∗ ) is a directed relational system with involutionwhich is a Kleene relational system, the mapping x ( x, a ) is an embedding of A into ( P a ( A ) , T ) .Proof. Let ( b, c ) , ( d, e ) ∈ P a ( A ).(i) Put B := ( P a ( A ) , T, ∗ ). From ( b, c ) ∈ P a ( A ) we conclude ( L ( b, c ) , a ) , ( a, U ( b, c )) ∈ R and hence ( L ( c, b ) , a ) , ( a, U ( c, b )) ∈ R , i.e. ( b, c ) ∗ = ( c, b ) ∈ P a ( A ) which shows that P a ( A ) is closed with respect to ∗ . According to Theorem 5.4, B is a directedrelational system with involution. Because of( L (( b, c ) , ( c, b )) , ( a, a )) = ( L ( b, c ) × U ( b, c ) , ( a, a )) ∈ S, (( a, a ) , U ( d, e ) × L ( d, e )) = (( a, a ) , U (( d, e ) , ( e, d ))) ∈ S we have ( L (( b, c ) , ( c, b )) , U (( d, e ) , ( e, d ))) ∈ S due to transitivity of S (which followsfrom the transitivity of R ) and hence B is a Kleene relational system.(ii) For all x ∈ A we have ( L ( x, a ) , a ) , ( a, U ( x, a )) ∈ R and hence ( x, a ) ∈ P a ( A ). Therest follows from Theorem 5.4.It should be remarked that if R is transitive then ( P a ( A ) , T, ∗ ) is a relational subsystemof the twist-product ( A , S, ∗ ) of A . References [1] G. Birkhoff, Lattice Theory. Amer. Math. Soc., Providence, R.I., 1979. ISBN 0-8218-1025-1.[2] S. Bonzio and I. Chajda, Residuated relational systems. Asian-Eur. J. Math. (2018), 1850024, 14pp.[3] I. Chajda, Sheffer operation in ortholattices. Acta Univ. Palack. Olomuc. Fac. RerumNatur. Math. (2005), 19–23.[4] I. Chajda and M. Kolaˇr´ık, Sheffer operations in complemented posets. Mathematicsfor Applications (to appear).[5] I. Chajda, M. Kolaˇr´ık and H. L¨anger, Algebras assigned to ternary relations, MiskolcMath. Notes (2013), 827–844.[6] I. Chajda and H. L¨anger, Groupoids assigned to relational systems, Math. Bohem. (2013), 15–23.[7] I. Chajda and H. L¨anger, Groupoids corresponding to relational systems. MiskolcMath. Notes (2016), 111–118.[8] I. Chajda and H. L¨anger, Relational systems with involution. Asian-Eur. J. Math. (2016), 1650087, 8pp.[9] I. Chajda and H. L¨anger, Kleene posets and pseudo-Kleene posets. Miskolc Math.Notes (submitted). http://arxiv.org/abs/2006.04417.1410] I. Chajda, H. L¨anger and P. ˇSevˇcik, An algebraic approach to binary relations.Asian-Eur. J. Math. (2015), 1550017, 13 pp.[11] R. Fraiss´e, Sur l’extension aux relations de quelques propri´et´es des ordres. Ann. Sci.Ecole Norm. Sup. (1954), 363–388.[12] J. A. Kalman, Lattices with involution. Trans. Amer. Math. Soc. (1958), 485–491.[13] J. Riguet, Relations binaires, fermetures, correspondances de Galois, Bull. Soc.Math. France (1948), 114–155.[14] H. M. Sheffer, A set of five independent postulates for Boolean algebras, with appli-cation to logical constants. Trans. Amer. Math. Soc.14