S-acts over a Well-ordered Monoid with Modular Congruence Lattice

This work relates to the structural act theory. The structural theory includes the description of acts over certain classes of monoids or having certain properties, for example, satisfying some requirement for the congruence lattice. The congruences of universal algebra is the same as the kernels of homomorphisms from this algebra into other algebras. Knowledge of all congruences implies the knowledge of all the homomorphic images of the algebra. A left $S$–act over monoid $S$ is a set $A$ upon which $S$ acts unitarily on the left. In this paper, we consider $S$–acts over linearly ordered and over well-ordered monoids, where a linearly ordered monoid $S$ is a linearly ordered set with a minimal element and with a binary operation $ \ max$, with respect to which $S$ is obviously a commutative monoid; a well-ordered monoid $S$ is a well-ordered set with a binary operation $ \ max$, with respect to which $S$ is also a commutative monoid. The paper is a continuation of the work of the author in co-authorship with M.S. Kazak, which describes $S$–acts over linearly ordered monoids with a linearly ordered congruence lattice and $S$-acts over a well-ordered monoid with distributive congruence lattice. In this article, we give the description of S-acts over a well-ordered monoid such that the corresponding congruence lattice is modular.

Congrunce lattices of rectangular lattices1

Let D be a finite distributive lattice with n join-irreducible elements. It is well-known that D can be represented as the congruence lattice of a rectangular lattice L which is a special planer semimodular lattice. In this paper, we shall give a better upper bound for the size of L by a function of n, improving a 2009 result of G. Grätzer and E. Knapp.

A MINIMAL CONGRUENCE LATTICE REPRESENTATION FOR

Let $p$ be an odd prime. The unary algebra consisting of the dihedral group of order $2p$, acting on itself by left translation, is a minimal congruence lattice representation of $\mathbb{M}_{p+1}$.

The largest higher commutator sequence

Given the congruence lattice L of a finite algebra A that generates a congruence permutable variety, we look for those sequences of operations on L that have the properties of higher commutator operations of expansions of A. If we introduce the order of such sequences in the natural way the question is whether exists or not the largest one. The answer is positive. We provide a description of the largest element and as a consequence we obtain that the sequences form a complete lattice.

Congruence lattices forcing nilpotency

Given a lattice [Formula: see text] and a class [Formula: see text] of algebraic structures, we say that [Formula: see text] forces nilpotency in [Formula: see text] if every algebra [Formula: see text] whose congruence lattice [Formula: see text] is isomorphic to [Formula: see text] is nilpotent. We describe congruence lattices that force nilpotency, supernilpotency or solvability for some classes of algebras. For this purpose, we investigate which commutator operations can exist on a given congruence lattice.

Characterization of congruence lattices of principal p-algebras

A characterization of the congruence lattice of a principal

Irredundant Decomposition of Algebras into One-Dimensional Factors

We introduce a notion of dimension of an algebraic lattice and, treating such a lattice as the congruence lattice of an algebra, we introduce the dimension of an algebra, too. We define a star-product as a special kind of subdirect product. We obtain the star-decomposition of algebras into one-dimensional factors, which generalizes the known decomposition theorems e.g. for Abelian groups, linear spaces, Boolean algebras.

Decomposition of Congruence Modular Algebras into Atomic, Atomless Locally Uniform and Anti-Uniform Parts

We describe here a special subdirect decomposition of algebras with modular congruence lattice. Such a decomposition (called a star-decomposition) is based on the properties of the congruence lattices of algebras. We consider four properties of lattices: atomic, atomless, locally uniform and anti-uniform. In effect, we describe a star-decomposition of a given algebra with modular congruence lattice into two or three parts associated to these properties.