On commutative unary algebras with the distributive congruations lattices

V. V. Popov

Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 21 (2018) no. 3, pp. 19-30.

Résumé

The article is devoted to the study of lattices of congruences of unary algebras. Algebras with $m$ unary operations were considered by A. I. Maltsev [6, p. 348] and were called $m$-unoids. Unar is an algebra with one unary operation. In [2; 3; 11] unars whose congruence lattices belong to a given class of lattices (semimodular, atomic, distributive, etc.) were studied. Similar questions for unary algebras with two unary operations were considered in [7; 8; 10]. Important results on commutative unary algebras with a distributive lattice congruences were obtained in [4; 5]. The main results of this note is announced in [9]. The unary algebra A$ = \left (A, \Omega \right)$ is an algebraic system, which is defined by some set $A$ and a set $\Omega$ of unary operations on $A$. Each operation $f\in\Omega$ can be considered as a mapping of the set $A$ into itself. The algebra A$ = \left (A,\Omega\right)$ is said to be commutative if for all $f, g \in \Omega$ and all $x\in A$ it holds the equality $f(g(x)) = g(f(x))$. The congruence $\theta$ on the algebra A is such an equivalence relation on $A$, that for each $f\in\Omega$ and all $x,y \in A$ from $x\theta y$ it follows $f(x)\theta f(y)$. By ${\rm Con} \ $Ais denoted the set of all congruences on algebra A. There is a partial order on ${\rm Con} \ $A: for the congruences $\theta_1, \theta_2$ the relation $\theta _1 \leq \theta _2$ is satisfied if and only if for any elements $x, y \in A$ from $x \theta_1 y$ it follows $x \theta_2 y $. If $\theta_1, \theta_2 \in {\rm Con} \ $A, then $\theta_1 \wedge \theta_2 $ denotes the lower bound congruences $\theta_1 $ and $\theta_2 $, then is the largest congruence $\theta \in {\rm Con} \ $A for which $\theta \leq \theta_1 $ and $\theta \leq \theta_2 $. The upper bound $\theta_1 \vee \theta_2 $ of congruences $\theta_1 $ and $\theta_2 $. A lattice of congruences ${\rm Con} \ $A is called distributive if for any three congruences $\theta_1, \theta_2, \theta_3 \in {\rm Con} \ $A the equality $\theta_1 \wedge (\theta_2 \vee \theta_3) = (\theta_1 \wedge \theta_2) \vee (\theta_1 \wedge \theta_3) $. Below we need the description of the following unars and unary algebras: Example 1 The unar D$_1$ is (N, $f$), where N is the set of natural numbers, and the operation $f $ is defined by the formula $f (x) = x + 1 $, $x \in $N . Example 2 For natural numbers $n\geq 1$, the unar D$_2$ is (Z$_n, f $), where Z$_n$ is the residue ring modulo $n$ and $f (x) = x + 1 ($mod$\ n) $ for $x \in $Z$_n $. If, in addition, $n = 1$, then the unary carrier consists of a single element, and $f$ is the identity map. Example 3 The unary algebra D$_3$ is (Z,$ f,g$), where Z is the set of integers, and $f$, $g$ are defined by formulas $f (x) = x + 1 $ and $g (x) = x-1 $, $x \in \mathbf{Z} $. The main result of this note is as follows: Theorem 1. Let A $= \left (A, \Omega \right) $ be a commutative unary algebra with a distributive lattice of congruences, $m = |\Omega| \geq 2 $. Then this algebra contains a subalgebra, the lattice of congruences of which is isomorphic to the lattice of congruences one of the unars D$_1 $, D$_2 (n) $ or a lattice congruences of the algebra D$_3 $.

Détail

Keywords: commutative unary algebra, latticies of congruences, distributive property
Mots-clés : $m$-unoid, cyclic element.

Bibliographie

[1] V. A. Artamonov, V. N. Saliy, L. A. Skornyakov, General Algebra, v. 1, Nauka Publ., Moscow, 1991, 480 pp.

[2] A. P. Boshchenko, “Pseudocomplementations in the Lattice of Congruences of Unary Algebras”, Algebraicheskie sistemy, mezhvuz. sb. nauchn. rabot, VGPI Publ., Volgograd, 1989, 23–26

[3] D. P. Egorova, “The Structure of Congruences of a Unary Algebra”, mezhvuz. nauch. sb., Uporyadochennye mnozhestva i reshetki, 5, Izd-vo Sarat. un-ta Publ., Saratov, 1978, 11–44

[4] V. K. Kartashov, A. V. Kartashova, V. N. Ponomarev, “On Conditions for Distributivity Or Modularity of Congruence Lattices of Commutative Unary Algebras”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 13:4, no. 2 (2013), 52–57

[5] A. V. Kartashova, “On Congruence Lattices of Direct Sums of Strongly Connected Commutative Unary Algebras”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 13:4, no. 2 (2013), 57–62

[6] A. I. Maltsev, Algebraic Systems, Nauka Publ., Moscow, 1970, 392 pp.

[7] V. V. Popov, On Collective Normality, on Rotated Graphs and Congruences of Unoids, LAMBERT Academic Publishing, Saarbrucken, 2013, 68 pp.

[8] V. V. Popov, “On Lattices of Congruences of Periodic Unary Algebras”, Science Journal of Volgograd State University. Mathematics. Physics, 2:21 (2014), 27–30

[9] V. V. Popov, “On the Distributivity of Lattices of Congruences of $m$-Unoids”, Natsionalnaya assotsiatsiya uchenykh, 7:34 (2017), 25–26

[10] V. L. Usoltsev, Minimal Unary Algebras with Two Commuting Operations, Dep. v VINITI 31.12.96 N3857-D96, 20 pp.

[11] J. Berman, “On the congruence lattices of unary algebras”, Proc. Amer. Math. Soc., 36:1 (1972), 34–38 | DOI