Abstract
In this paper we consider the variety V P of algebras with one unary and one ternary operation p that satisfies the Pixley identities, provided that operations are permutable. We describe the structure of a free algebra of the variety V P and study the structure of unary reducts of free algebras. We prove the solvability of the word problem in free algebras and the uniqueness of a free basis; we also describe groups of automorphisms of free algebras. Similar results are obtained for free algebras of a subvariety of the variety V P defined by the identities p(p(x, y, z), y, z) = p(x, y, z) and p(x, y, p(x, y, z)) = p(x, y, z).
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References
R. Freese and R. McKenzie, Commutator Theory for Congruence Modular Varieties (Lond. Math Soc. Lect. Notes Series, 1987), Vol. 125.
A. P. Zamyatin, Varieties with Restrictions on the Lattice of Congruences (Ural’sk. Univ., Sverdlovsk, 1987) [in Russian].
A. I. Mal’tsev, “On the General Theory of Algebraic Systems,” Matem. Sborn. (New Series) 35(77), 3–20 (1954).
A. F. Pixley, “Distributivity and Permutability of Congruence Relations in Equational Classes of Algebras,” Proc. Amer. Math. Soc. 14(1), 105–109 (1963).
L. A. Skornjakov, “Unars,” Coll.Math. Soc. J. Bolyai, No. 29, 735–743 (1977).
V. K. Kartashov, “Quasivarieties of Unars,” Matem. Zametki 27(1), 7–20 (1980).
V. K. Kartashov, “About Lattices of Quasivarieties of Unars” Sib. Matem. Zhurn. 26(3), 49–62 (1985).
V. K. Kartashov, “Quasivarieties of Unars with Finite Number of Cycles,” Algebra i Logika 19(2), 173–193 (1980).
A. I. Mal’tsev, Algebraic Systems (Nauka, Moscow, 1970) [in Russian].
M. Kilp, U. Knauer, A. V. Mikhalev, and L. A. Skornjakov, Acts over Monoids. A Bibliographical Survey of Publications 1975–1981 (Verlag, Oldenburg, 1982).
M. Kilp, U. Knauer, A. V. Mikhalev, Monoids, Acts and Categories (Walter de Gruyter, Berlin, 2000).
D.M. Smirnov, “Correspondence between Regular Varieties of Unary Algebras and Semigroups,” Algebra i Logika 17(4), 468–477 (1978).
I. B. Kozhukhov, “Semigroups, over which All Acts are Residually Finite,” Fundament. i Prikl.Matem. 4(4), 1335–1344 (1998).
V. K. Kartashov, “About Unars with Mal’tsev’s Operation,” in Abstract Algebra and its Applications: International seminar dedicated to thememory of Prof. L. A. Skornyakov (Volgograd, 1999), pp. 31–32.
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Original Russian Text © V.L. Usol’tsev, 2009, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2009, No. 4, pp. 43–49.
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Usol’tsev, V.L. Free algebras of a unary variety with Mal’tsev’s operation that satisfies the Pixley conditions. Russ Math. 53, 34–39 (2009). https://doi.org/10.3103/S1066369X09040069
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DOI: https://doi.org/10.3103/S1066369X09040069