Abstract
In this paper, we obtain axiom systems, bases of identities and quasi-identities for classes of algebras of binary relations with a conjunctive operation, i.e., an operation that can be defined by a logical formula containing only conjunctions. The result of applying such an operation to two binary relations is the Cartesian product of their reflexive projections. Our consideration of these classes leads to the concept of quasi-semilattices as a natural generalization of the notion of semilattices.
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Andréka, H., and D. A. Bredikhin, The equational theory of union-free algebras of relations, Algebra Universalis 33:516–532, 1995.
Andréka, H., I. Nemeti, and I. Sain, Algebraic Logic, in D.M. Gabbay, and F. Guenthner, (eds.), Handbook of Philosophical Logic, vol. 2, 2nd ed., Kluwer Academic Publishers, 2001, pp. 133–247.
Boner, P., and F. R. Poschel, Clones of operations on binary relations, Contributions to general algebra 7:50–70, 1991.
Bredikhin, D. A., On relation algebras with general superpositions, in H. Andreka, J. Monk, and I. Nemeti, (eds.), Algebraic Logic, North-Holland, Amsterdam, 1991, pp. 111–124.
Bredikhin, D. A., Varieties of groupoids associated with involuted restrictive bisemigroups of binary relations, Semigroup Forum 44:87–192, 1992.
Bredikhin, D. A., On algebras of relations, Banach Center Publications 28:191–199, 1992.
Bredikhin, D. A., The equational theory of algebras of relations with positive operations, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika 3:23–30, 1993 (Russian).
Bredikhin, D. A., On quasi-identities of algebras of relations with Diophantine operations, Siberian Mathematical Journal 38:23–33, 1997.
Bredikhin, D. A., On Varieties of Groupoids of Relations with Operation of Binary Cylindrification, Algebra Universalis 73:43–52, 2015.
Bredikhin, D. A., On partially ordered semigroups of relations with domino operations, Semigroup Forum 92:198–213, 2016.
Bredikhin, D. A., On generalized subreducts of Tarski’s algebras of relations with the operation of bi-directional intersection, Algebra Universalis 79:77–92, 2018.
Bredikhin, D. A., On Classes of Generalized Subreducts of Tarski’s Relation Algebras with One Diophantine Binary Operation, Mathematical Notes 106:872–884, 2019.
Bredikhin, D. A., On semigroups of relations with primitive-positive operations of rank two, Mathematics. Mechanics, Saratov State University Publishing House 21:10–11, 2019.
Bredikhin, D. A., On Jonsson’s Problem for Groupoids of Relations with Operation of Binary Cylindrification, Algebra Universalis 82:Article number 3, 2021.
Bredikhin, D. A., On Algebras of Binary Relations with Conjunctive Operations, Algebra Universalis 82:Article number 39, 2021.
Hirsch, R., and S. Mikulas, Ordered domain algebras, Journal of Applied Logic 11:266–271, 2013.
Hodkinson, I., and S. Mikulas, Axiomatizability of reducts of algebras of relations, Algebra Universalis 43:127–156, 2000.
Jonsson, B., Representation of modular lattices and of relation algebras, Transactions of the American Mathematical Society 92:449–464, 1959.
Lyndon, R. C., The representation of relation algebras, II, Annals of Mathematics 63:294–307, 1956.
Markov, V. T., A. V. Mikhalev, and A. A. Nechaev, Nonassociative algebraic structures in cryptography and coding, Fundamentalnaya i prikladnaya matematika 4:99–123, 2016.
Schein, B. M., Relation algebras and function semigroups, Semigroup Forum 1:1–62, 1970.
Schein, B. M., Representation of subreducts of Tarski relation algebras, in H. Andreka, J. Monk, and I. Nemeti, (eds.), Algebraic Logic, North-Holland, Amsterdam, 1991, pp. 621–635.
Tarski, A., On the calculus of relations, The Journal of Symbolic Logic 6:73–89, 1941.
Tarski, A., Contributions to the theory of models, III, Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 58:56–64, 1955.
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Presented by Yde Venema; Received February 24, 2021.
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Bredikhin, D. On Groupoids of Relations with One Conjunctive Operation of Rank 2. Stud Logica 110, 1137–1153 (2022). https://doi.org/10.1007/s11225-022-09993-2
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DOI: https://doi.org/10.1007/s11225-022-09993-2