Semilattice Indecomposable Finite Semigroups With Large Subsemilattices

TL;DR

This paper investigates the size limitations of subsemilattices within finite semilattice indecomposable semigroups, characterizes those with specific sizes, and introduces new algebraic characterizations based on semigroup algebra properties.

Contribution

It establishes an upper bound on subsemilattice size, characterizes certain semigroups with maximal subsemilattices, and introduces a novel algebraic approach for their analysis.

Findings

Maximum subsemilattice size is bounded by a function of the semigroup size.

Semigroups with maximal subsemilattices are identified as special inverse semigroups.

A new algebraic characterization for semilattice indecomposable semigroups with zero is provided.

Abstract

In this paper we show that if $Y$ is a subsemilattice of a finite semilattice indecomposable semigroup $S$ then $|Y|\leq 2\left\lfloor \frac{|S|-1}{4}\right\rfloor+1$. We also characterize finite semilattice indecomposable semigroups $S$ which contains a subsemilattice $Y$ with $|S|=4k+1$ and $|Y|=2\left\lfloor \frac{|S|-1}{4}\right\rfloor+1=2k+1$. They are special inverse semigroups. Our investigation is based on our new result proved in this paper which characterize finite semilattice indecomposable semigroups with a zero by only use the properties of its semigroup algebra.