Sheffer operation in relational systems

Abstract

The concept of a Sheffer operation known for Boolean algebras and orthomodular lattices is extended to arbitrary directed relational systems with involution. It is proved that to every such relational system, there can be assigned a Sheffer groupoid and also, conversely, every Sheffer groupoid induces a directed relational system with involution. Hence, investigations of these relational systems can be transformed to the treatment of special groupoids which form a variety of algebras. If the Sheffer operation is also commutative, then the induced binary relation is antisymmetric. Moreover, commutative Sheffer groupoids form a congruence distributive variety. We characterize symmetry, antisymmetry and transitivity of binary relations by identities and quasi-identities satisfied by an assigned Sheffer operation. The concepts of twist products of relational systems and of Kleene relational systems are introduced. We prove that every directed relational system can be embedded into a directed relational system with involution via the twist product construction. If the relation in question is even transitive, then the directed relational system can be embedded into a Kleene relational system. Any Sheffer operation assigned to a directed relational system $${\mathbf {A}}$$ A with involution induces a Sheffer operation assigned to the twist product of $${\mathbf {A}}$$ A .

Class of Sheffer stroke BCK-algebras

“…So, many researchers want to use this operation on their studies. For example, Sheffer stroke non-associative MV-algebras [3] and filters [13], (fuzzy) filters of Sheffer stroke BL-algebras [14], Sheffer stroke Hilbert algebras [11] and filters [12], Sheffer stroke UP-algebras [15], Sheffer stroke BG-algebras [16], Sheffer stroke BE-algebras [17] and Sheffer operation in ortholattices [2] are given as some research on Sheffer stroke operation in recent years.…”

“…[2] Let A = (A, |) be a groupoid. The operation | is said to be Sheffer stroke if it satisfies the following conditions:(S1) x|y = y|x, (S2) (x|x)|(x|y) = x,(S3) x|((y|z)|(y|z)) = ((x|y)|(x|y))|z, (S4) (x|((x|x)|(y|y)))|(x|((x|x)|(y|y))) = x. Lemma 2.1.…”

Sheffer operation in relational systems

“…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 only in Boolean algebras but also in orthomodular lattices or even in ortholattices (see [1] for these concepts). These algebras form an algebraic axiomatization of the logic of quantum mechanics.…”