Semigroups through semilattices

Abstract:

Presented in this paper is a method of constructing a compact semigroup $S$ from a compact semilattice $X$ and a compact semigroup $T$ having idempotents contained in $X$. The notions of semigroups (straight) through chains and (straight) through semilattices are introduced. It is shown that the notion of a semigroup through a chain is equivalent to that of a generalized hormos. Universal objects are obtained in several categories including the category of clans straight through a chain and the category of clans straight through a semilattice relative to a chain. An example is given of a nonabelian clan $S$ with abelian set of idempotents $E$ such that $S$ is minimal (as a clan) about $E$.