Narrow Results

Hedrlín and Pultr proved that for any monoid M there exists a graph G with endomorphism monoid isomorphic to M. In a previous paper, we give a construction G(M) for a graph with prescribed endomorphism monoid M known as a -graph. Using this construction, we derived bounds on the minimum number of vertices and edges required to produce a graph with a given endomorphism monoid for various classes of finite monoids. In this paper, we generalize the -graph construction and derive several new bounds for monoid classes not handled by our first paper. Among these are the so called "strong semilattices of C-semigroups" where C is one of the following: Groups, Abelian Groups, Rectangular Groups, and completely simple semigroups.

For $n$ a positive integer and $K$ a finite set of finite algebras, let $L(n,K)$ denote the largest $n$-generated subdirect product whose subdirect factors are algebras in $K$. When $K$ is the set of all $n$-generated subdirectly irreducible algebras in a locally finite variety ${𝒱}$, then $L(n,K)$ is the free algebra $F_{{𝒱}}(n)$ on $n$ free generators for ${𝒱}$. For a finite algebra $A$ the algebra $L(n,\{A\})$ is the largest $n$-generated subdirect power of $A$. For every $n$ and finite $A$ we provide an upper bound on the cardinality of $L(n,\{A\})$. This upper bound depends only on $n$ and these basic parameters: the cardinality of the automorphism group of $A$, the cardinalities of the subalgebras of $A$, and the cardinalities of the equivalence classes of certain equivalence relations arising from congruence relations of $A$. Using this upper bound on $n$-generated subdirect powers of $A$, as $A$ ranges over the $n$-generated subdirectly irreducible algebras in ${𝒱}$, we obtain an upper bound on $|F_{{𝒱}}(n)|$. And if all the $n$-generated subdirectly irreducible algebras in ${𝒱}$ have congruence lattices that are chains, then we characterize in several ways those ${𝒱}$ for which this upper bound is obtained.

A finite unary algebra $(A,F)$ has only countably many countable subdirect powers if and only if every operation $f\in F$ is either a permutation or a constant mapping.