Sheffer operation in relational systems

TL;DR

This paper extends the Sheffer operation concept to directed relational systems with involution, establishing a correspondence with Sheffer groupoids and exploring their algebraic properties and embeddings.

Contribution

It introduces Sheffer operations for arbitrary directed relational systems, characterizes their properties, and connects them to algebraic structures like groupoids and relational systems.

Findings

Sheffer groupoids form a variety of algebras.

Commutative Sheffer groupoids form a congruence distributive variety.

Directed relational systems can be embedded into Kleene 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 treaty 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 symmertry, antisymmetry and treansitivity 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 A with involution induces a Sheffer operation assigned to the twist-product of A.