Geodesic
On congruence lattices of algebras with an operator and the symmetric main operation
Čebyševskij sbornik, Tome 25 (2024) no. 1, pp. 103-115.

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In this paper we study properties of congruence lattices of algebras with one operator and the main symmetric operation. A ternary operation $d(x,y,z)$ satisfying identities $d(x, y, y) = d(y, y, x) = d(y, x, y) = x$ is called a minority operation. The symmetric operation is a minority operation defined by specific way. An algebra $A$ is called a chain algebra if $A$ has a linearly ordered congruence lattice. An algebra $A$ is called subdirectly irreducible if $A$ has the smallest nonzero congruence. An algebra with operators is an universal algebra whose signature consists of two nonempty non-intersectional parts: the main part which can contain arbitrary operations, and the additional part consisting of operators. The operators are unary operations that act as endomorphisms with respect to the main operations, i.e., one are permutable with the main operations. An unar is an algebra with one unary operation. If $f$ is the unary operation from the signature $\Omega$ then the unar $\langle A, f\rangle$ is called an unary reduct of algebra $\langle A, \Omega\rangle$. A description of algebras with one operator and the main symmetric operation that have a linear ordered congruence lattice is obtained. It shown that algebra of given class is a chain algebra if and only if one is subdirectly irreducible. For algebras of given class we obtained necessary and sufficient conditions for the coincidence of their congruence lattices and congruence lattices of unary reducts these algebras.
Keywords: congruence lattice, algebra with operators, unary reduct of algebra, chain algebra, subdirectly irreducible algebra. 
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V. L. Usoltsev. On congruence lattices of algebras with an operator and the symmetric main operation. Čebyševskij sbornik, Tome 25 (2024) no. 1, pp. 103-115. http://geodesic.mathdoc.fr/item/CHEB_2024_25_1_a6/

[1] Tamura T., “Commutative semigroups whose lattice of congruences is a chain”, Bull. Soc. Math. France, 97 (1969), 369–380 | DOI | MR | Zbl

[2] Schein B. M., “Commutative semigroups where congruences form a chain”, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 17 (1969), 523–527 | MR | Zbl

[3] Schein B. M., “Corrigenda to “Commutative semigroups where congruences form a chain””, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 23 (1975), 1247–1248 | MR

[4] Nagy A., Jones P. R., “Permutative Semigroups Whose Congruences Form a Chain”, Semigroup Forum, 69 (2004), 446–456 | DOI | MR | Zbl

[5] Kozhukhov I. B., “Left chain semigroups”, Semigroup Forum, 22 (1981), 1–8 | DOI | MR

[6] Popovich A. L., Jones P. R., “On congruence lattices of nilsemigroups”, Semigroup Forum, 95:2 (2017), 314–320 | DOI | MR | Zbl

[7] Goldberg M. S., “Distributive double p-algebras whose congruence lattices are chains”, Algebra Universalis, 17 (1983), 208–215 | DOI | MR | Zbl

[8] Egorova, D. P., “The congruence lattice of a unary algebra”, Mejvuzovskii nauchnii sbornik, Ordered sets and lattices, 5, Saratov, 1978, 11–44 (Russian)

[9] Kartashova, A. V., “On commutative unary algebras with totally ordered congruence lattice”, Mathematical Notes, 95:1 (2014), 67–77 | DOI | DOI | MR | Zbl

[10] Szendrei A., Clones in universal algebra, Les presses de l'Université de Montréal, Montréal, 1986, 166 pp. | MR | Zbl

[11] Hyndman J., Nation J. B., Nishida J., “Congruence lattices of semilattices with operators”, Studia Logica, 104:2 (2016), 305–316 | DOI | MR | Zbl

[12] Garcia P., Esteva F., “On Ockham Algebras: Congruence Lattices and Subdirectly Irreducible Algebras”, Studia Logica, 55 (1995), 319–346 | DOI | MR | Zbl

[13] Kartashov, V. K., “On unars with Mal'tsev operation”, Universal algebra and application, theses of Int. workshop dedicated memory of prof. L. A. Skornyakov, Volgograd, 1999, 31–32 (Russian)

[14] Usoltsev, V. L., “On Rees closure in some classes of algebras with an operator”, Chebyshevskiy sbornik, 22:2(78) (2021), 271–287 (Russian) | DOI | DOI | MR | Zbl

[15] Usoltsev, V. L., “Unars with ternary Mal'tsev operation”, Russian Mathematical Surveys, 63:5 (2008), 986–987 | DOI | DOI | MR | Zbl

[16] Usoltsev, V. L., “On subdirectly irreducible unars with Mal'tsev operation”, Izvestiya Volgogradskokgo gosudarstvennogo pedagogicheskogo universiteta. Seriya “Estestvennye i fiziko-matematicheskie nauki”, 2005, no. 4(13), 17–24 (Russian)

[17] Usoltsev, V. L., “On polynomially complete and Abelian unars with Mal'tsev operation”, Uchenye Zapiski Orlovskogo Gosudarstvennogo Universiteta, 6(50):2 (2012), 229–236 (Russian)

[18] Usoltsev, V. L., “On Hamiltonian ternary algebras with operators”, Chebyshevskiy sbornik, 15:3(51) (2014), 100–113 (Russian) | DOI | MR | Zbl

[19] Maróti M., McKenzie R., “Existence theorems for weakly symmetric operations”, Algebra Universalis, 59:3-4 (2008), 463–489 | DOI | MR | Zbl

[20] Bulatov A., Krokhin A., Jeavons P., “The complexity of constraint satisfaction: An algebraic approach”, Structural Theory of Automata, Semigroups and Universal Algebra, Springer-Verlag, Berlin, 2005, 181–213 | DOI | MR

[21] Baker K. A., Pixley A., “Polynomial interpolation and the Chinese Remainder Theorem for algebraic systems”, Math. Zeitschrift, 143 (1975), 165–174 | DOI | MR | Zbl

[22] Marković P., McKenzie R., “Few subpowers, congruence distributivity and near-unanimity terms”, Algebra Universalis, 58 (2008), 119–128 | DOI | MR | Zbl

[23] Usol'tsev, V. L., “Subdirectly Irreducible Algebras in One Class of Algebras with One Operator and the Main Near-Unanimity Operation”, Lobachevskii Journal of Mathematics, 42:1 (2021), 206–216 | DOI | MR | Zbl

[24] Usoltsev, V. L., “On congruence lattices of algebras with one operator and basic near-unanimity operation”, Nauchno-tekhnicheskiy vestnik Povolzhya, 2016, no. 2, 28–30 (Russian)

[25] Wenzel G. H., “Subdirect irreducibility and equational compactness in unary algebras $\langle A; f \rangle$”, Arch. Math. (Basel), 21 (1970), 256–264 | DOI | MR | Zbl

[26] Lata, A. N., “On congruence coherent Rees algebras and algebras with an operator”, Chebyshevskiy sbornik, 18:2(62) (2017), 154–172 (Russian) | DOI | MR | Zbl

[27] Usoltsev, V. L., “The subdirect irreducibility and the atoms of congruence lattices of algebras with one operator and the symmetric main operation”, Chebyshevskiy sbornik, 22:2(78) (2021), 257–270 (Russian) | DOI | MR | Zbl

[28] Usoltsev V. L., “Simple and pseudosimple algebras with operators”, Journal of Mathematical Sciences, 164:2 (2010), 281–293 | DOI | MR | Zbl