Abstract
In the paper, systems of elementary axioms are found for classes of groupoids and ordered groupoids of binary relations with Diophantine operations that are generalized subreducts of the Tarski relation algebras.
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H. Andréka, I. Neméti, and I. Sain, “Algebraic logic,” in Handbook of Philosophical Logic, Vol. 2 (Kluwer Acad. Publ., Dordrecht, 2001), pp. 133–247.
A. Tarski, “On the calculus of relations,” J. Symbolic Logic 6, 73–89 (1941).
A. Tarski, “Contributions to the theory of models. III,” Nederl. Akad. Wetensch. Proc. Ser. A 58, 56–64 (1955).
R. C. Lyndon, “The representation of relation algebras. II,” Ann. of Math. (2) 63, 294–307 (1956).
J. D. Monk, “On representable relation algebras,” Michigan Math. J. 11, 207–210 (1964).
B. Jónsson, “Representation of modular lattices and of relation algebras,” Trans. Amer. Math. Soc. 11, 449–464 (1959).
M. Haiman, “Arguesian lattices which are not type 1,” Algebra Universalis 28 (1), 128–137 (1991).
H. Andréka and D. A. Bredikhin, “The equational theory of union-free algebras of relations,” Algebra Universalis 33 (4), 516–532 (1994).
B. Jónsson, “The theory of binary relations,” in Algebraic Logic, Colloq. Math. Soc. János Bolyai (North-Holland, Amsterdam, 1991), Vol. 54, pp. 245–292.
D. A. Bredikhin, “On quasi-identities of relation algebras with Diophantine operations,” Sibirsk. Mat. Zh. 38 (1), 29–41 (1997) [SiberianMath. J. 38 (1), 23–33 (1997)].
D. A. Bredikhin, “Relation algebras with Diophantine operations,” Dokl. Ross. Akad. Nauk 360 (5), 594–595 (1998) [Dokl. Math. 57 (3), 435–436 (1998)].
F. Bönere and R. Pöschel, “Clones of operations on binary relations,” in Contributions to General Algebra, 7 (Hölder-Pichler-Tempsky, Vienna, 1991), pp. 50–70.
D. A. Bredikhin, “On relation algebras with general superpositions,” in Algebraic Logic, Colloq. Math. Soc. János Bolyai (North-Holland, Amsterdam, 1991), Vol. 54, pp. 111–124.
V. V. Vagner, “Restrictive semigroups,” Izv. Vyssh. Uchebn. Zaved. Mat. No. 6, 19–27 (1962) [in Russian].
K. A. Zaretskii, “The representation of ordered semigroups by binary relations,” Izv. Vyssh. Uchebn. Zaved. Mat. No. 6, 48–50 (1959) [in Russian].
H. Andréka, “Representation of distributive lattice-ordered semigroups with binary relations,” Algebra Universalis 28 (1), 12–25 (1991).
H. Andréka and Sz. Mikulás, “Axiomatizability of positive algebras of binary relations,” Algebra Universalis 66 (1–2), 7–34 (2011).
I. Hodkinson and Sz. Mikulás, “Axiomatizability of reducts of algebras of relations,” Algebra Universalis 43 (2–3), 127–156 (2000).
I. Hodkinson and Sz. Mikulás, “Representable semilattice-ordered monoids,” Algebra Universalis 57 (3), 333–370 (2007).
R. Hirsch and Sz. Mikulás, “Axiomatizability of representable domain algebras,” J. Log. Algebr. Program. 80 (2), 75–91 (2011).
I. Hodkinson and Sz. Mikulás, “Ordered domain algebras,” J. Appl. Log. 11 (3), 266–271 (2013).
B.M. Schein, “Relation algebras and function semigroups,” Semigroup Forum 1 (1), 1–62 (1970).
B.M. Schein, “Representation of subreducts of Tarski relation algebras,” in Algebraic Logic, Colloq.Math. Soc. János Bolyai (North-Holland, Amsterdam, 1991), Vol. 54, pp. 621–635.
D. A. Bredikhin, “On varieties of groupoids of relations with operation of binary cylindrification,” Algebra Universalis 73 (1), 43–52 (2015).
D. A. Bredikhin, “Identities of groupoids of relations with operation of cylindered intersection,” Izv. Vyssh. Uchebn. Zaved. Mat. No. 8, 12–20 (2018) [RussianMath. (Iz. VUZ) 62 (8), 9–16 (2018)].
D. A. Bredikhin, “On generalized subreducts of Tarski’s algebras of relations with the operation of bi-directional intersection,” Algebra Universalis 79 (Art. 77) (2018).
D. A. Bredikhin, “The equational theory of relation algebras with positive operations,” Izv. Vyssh. Uchebn. Zaved. Mat. No. 3, 23–30 (1993) [RussianMath. (Iz. VUZ) 37 (3), 21–28 (1993)].
J. D. Phillips, “Short equational bases for two varieties of groupoids associated with involuted restrictive bisemigroups of binary relations,” Semigroup Forum 73 (2), 308–312 (2006).
S. Yu. Katyshev, V. T. Markov, and A. A. Nechaev, “Application of non-associative groupoids to the realization of an open key distribution procedure,” Diskr. Mat. 26 (3), 45–64 (2014) [DiscreteMath. Appl. 25 (1), 9–24 (2015)].
L. Henkin, J. D. Monk, and A. Tarski, Cylindric Algebras, Part I (North-Holland Publishing, Amsterdam, 1971).
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Russian Text © The Author(s), 2019, published in Matematicheskie Zametki, 2019, Vol. 106, No. 6, pp. 821–836.
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Bredikhin, D.A. On Classes of Generalized Subreducts of Tarski’s Relation Algebras with One Diophantine Binary Operation. Math Notes 106, 872–884 (2019). https://doi.org/10.1134/S0001434619110221
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DOI: https://doi.org/10.1134/S0001434619110221