$\mathscr {P}$-Condense and $\mathscr {C}\mathscr {P}$-Condensing Operators on Semilattices, Join-Complete Lattices and Some Inverse Semigroups

Abstract

In this paper, the concepts of $\mathscr {P}$-condensing operator and $\mathscr {C}\mathscr {P}$-condensing operator on a semigroup are introduced. Then we describe the $\mathscr {P}$-condensing operators and the $\mathscr {C}\mathscr {P}$-condensing operators on semilattices, join-complete lattices, and inverse semigroups. As an application of our results, we also show that the unary operations${}^\dag $: $Hg\mapsto H\wedge g^{-1}Hg$ and ${}^\ddag $: $Hg\mapsto H_G$ on the coset semigroup $\mathbb {K}(G)$ of a group G are $\mathscr {P}$-condensing operators. Consequently, a necessary and sufficient condition for these operators to be $\mathscr {C}\mathscr {P}$-condensing operators is given. Some further results on condense semigroups considered by Chen et al. (Acta Math. Hung. 119: 281–305, 2008) are discussed and obtained in this paper.

Keywords

$\mathscr {P}$-condensing operator')">$\mathscr {P}$-condensing operator / $\mathscr {C}\mathscr {P}$-condensing operator')">$\mathscr {C}\mathscr {P}$-condensing operator / Upper closure operator / Join-complete congruence / Join-complete lattice congruence / Coset semigroup of a group