Abstract
О полугруппах отношений с операцией левого и правого прямоугольного произведения
Highlights
Let Rel(U ) be the set of all binary relations on a base set U
An operation on relations is called primitive-positive if it can be defined by a formula of the first-order predicate calculus containing only existential quantifiers and conjunctions in its prenex normal form
An operation on relations is called primitive-positive [2] if it can be defined by a formula of the first-order predicate calculus containing only existential quantifiers and conjunctions in its prenex normal form
Summary
Let Rel(U ) be the set of all binary relations on a base set U. Any algebra of relations with primitive-positive operations (Φ, Ω) can be considered as partially ordered (Φ, Ω, ⊆). He considered algebras of relations (Tarski’s algebras of relations) with the following operations: Boolean operations ∪, ∩, −; operations of relational product ◦ and relational inverse −1; constant operations ∆ (diagonal relation), ∅ (empty relation), ∇ = U × U (universal relation) He showed that the class R{◦, −1, ∪, ∩, −, ∆, ∅, ∇} is not a quasi-variety and the quasi-variety generated by this class forms a variety [7]. The classes R{⊲} and R{⊳} coincide with the following varieties of semigroups given by the identities x2 = x, xyz = yxz and x2 = x, xyz = xzy respectively. It will be sufficient to study only one of these operations
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More From: Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics
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