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}} with involution induces a Sheffer operation assigned to the twist product of {mathbf {A}}.
Highlights
Relational systems form one of the most general mathematical structures
At first, we show that as for Boolean algebras where all the operations can be recovered by means of the Sheffer operation, here the unary operation and the given binary relation can be reconstructed by means of only one specific Sheffer operation
Theorem 3.7 shows that if we start with a directed relational system A with involution, we consider a Sheffer groupoid G assigned to A, and we construct the directed relational system B with involution induced by G, B = A
Summary
Relational systems form one of the most general mathematical structures. Almost all structures appearing in algebra can be considered as relational structures. The technology of production of such chips is much easier and cheaper than it was in the beginning of computer era when several parts of the computer were composed by at least 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 Chajda (2005), a Sheffer operation can be introduced in Boolean algebras and in orthomodular lattices or even in ortholattices (see Birkhoff 1979 for these concepts). These algebras form an algebraic semantics of the logic of quantum mechanics, see e.g., Birkhoff and von Neumann (1936) or Husimi (1937). We derive Kleene relational systems by using the twist product construction and introduce a Sheffer operation on them in order to be able to apply the above-mentioned tools and results
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