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 Izv. Saratov Univ. Math. Mech. Inform., 2020, Volume 20, Issue 3, Pages 280–289 (Mi isu847)

Scientific Part
Mathematics

On semigroups of relations with the operation of left and right rectangular products

D. A. Bredikhin

Yuri Gagarin State Technical University of Saratov, 77 Politechnicheskaya St., Saratov 410054, Russia

Abstract: A set of binary relations closed with respect to some collection of operations on relations forms an algebra called an algebra of relations. The class of all algebras (partially ordered algebras) isomorphic to algebras (partially ordered by set-theoretic inclusion $\subseteq$ algebras) of relations with operations from $\Omega$ is denoted by $\mathrm{R}\{\Omega\}$ ($R\{\Omega,\subseteq\}$). 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. We consider algebras of relations with associative primitive-positive operations $\ast$ and $\star$, defined by the following formulas $\rho\ast\sigma=\{(u,v): (\exists s,t,w) (u,s)\in \rho \wedge (t,w)\in \sigma\}$ and $\rho\star\sigma=\{(u,v): (\exists s,t,w) (s,t)\in \rho \wedge (w,v)\in \sigma\}$ respectively. The axiom systems for the classes $\mathrm{R}\{\ast\}$, $\mathrm{R}\{\ast,\subseteq\}$, $\mathrm{R}\{\star\}$, $\mathrm{R}\{\star,\subseteq\}$, and bases of quasi-identities and identities for quasi-varieties and varieties generated by these classes are found.

Key words: algebra of relations, primitive positive operation, identity, variety, quasi-identity, quasi-variety, semigroup, partially ordered semigroup.

DOI: https://doi.org/10.18500/1816-9791-2020-20-3-280-289

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Bibliographic databases:

UDC: 501.1
Revised: 28.06.2019
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Citation: D. A. Bredikhin, “On semigroups of relations with the operation of left and right rectangular products”, Izv. Saratov Univ. Math. Mech. Inform., 20:3 (2020), 280–289

Citation in format AMSBIB
\Bibitem{Bre20} \by D.~A.~Bredikhin \paper On semigroups of relations with the operation of left and right rectangular products \jour Izv. Saratov Univ. Math. Mech. Inform. \yr 2020 \vol 20 \issue 3 \pages 280--289 \mathnet{http://mi.mathnet.ru/isu847} \crossref{https://doi.org/10.18500/1816-9791-2020-20-3-280-289} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000565963700001}