Semilattice Indecomposable Finite Semigroups With Large Subsemilattices

Abstract

In this paper we show that if $Y$ is a subsemilattice of a finite semilattice indecomposable semigroup $S$ then $|Y|\le 2\lfloor \frac{|S|-1}{4}\rfloor +1$. We also characterize finite semilattice indecomposable semigroups $S$ which contains a subsemilattice $Y$ with $|S|=4k+1$ and $|Y|=2\lfloor \frac{|S|-1}{4}\rfloor +1=2k+1$. 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.

11 pages