On coatoms and complements in congruence lattices of unars with Mal'tsev operation

A. N. Lata

Čebyševskij sbornik, Tome 16 (2015) no. 4, pp. 212-226.

Résumé

One important problem is studying of lattices that naturally associated with universal algebra. In this article is considered algebras $\langle A, p, f \rangle$ with one Mal'tsev operation $p$ and one unary operation $f$ acting as endomorphism with respect to operation $p$. We study properties of congruence lattices of algebras $\langle A, p, f \rangle$ with Mal’tsev operation $p$ that introduced by V. K. Kartashov. This algebra is defined as follows. Let $\langle A, f \rangle$ be an arbitrary unar and $x, y \in A$. For any element $x$ of the unar $\langle A, f \rangle $ by $f^n(x)$ we denote the result of $f$ applied $n$ times to an element $x$. Also $f^0(x)=x$. Assume that $$M_{x, y} = \{ n\in \mathbb{N} \cup \{0\} \mid f^{n}(x) = f^{n}(y) \}$$ and also $k(x, y) = \min M_{x, y}$, if $M_{x, y} \ne \emptyset$ and $k(x, y) = \infty$, if $M_{x, y} = \emptyset$. Assume further $$ p( x, y, z ) \stackrel{def}{=} \begin{cases} z, \text{ если } k(x,y) \leqslant k(y,z)\\ x, \text{ если } k(x,y) > k(y,z). \end{cases} $$ It is described a structure of coatoms in congruence lattices of algebras $\langle A, p, f \rangle$ from this class. It is proved congruence lattices of algebras $\langle A, p, f \rangle$ has no coatoms if and only if the unar $\langle A, f \rangle$ is connected, contains one-element subunar and has infinite depth. In other cases congruence lattices of algebras $\langle A, p, f \rangle$ has uniquely coatom. It is showed for any congruences $\theta \ne A \times A$ and $\varphi \ne A \times A$ of algebra $\langle A, p, f \rangle$ fulfills $\theta \vee \varphi A \times A$. Necessary and sufficient conditions when a congruence lattice of algebras from given class is complemented, uniquely complemented, relatively complemented, Boolean, generalized Boolean, geometric are obtained. It is showed any non-trivial congruence of algebra $\langle A, p, f \rangle$ from this class has no complement. It is proved that congruence lattices of any algebra $\langle A, p, f \rangle$ from given class is dual pseudocomplemented lattice. Bibliography: 24 titles.

Détail

Keywords: congruence lattice, complemented lattice, dual pseudocomplemented lattice, coatom (dual atom), algebra with operators, unar with Mal'tsev operation.

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