$\mathscr {P}$-condensing operator,$\mathscr {C}\mathscr {P}$-condensing operator,Upper closure operator,Join-complete congruence,Join-complete lattice congruence,Coset semigroup of a group" /> $\mathscr {P}$-condensing operator" /> $\mathscr {C}\mathscr {P}$-condensing operator" /> $\mathscr {P}$-condensing operator,$\mathscr {C}\mathscr {P}$-condensing operator,Upper closure operator,Join-complete congruence,Join-complete lattice congruence,Coset semigroup of a group" />
$\mathscr {P}$-Condense and $\mathscr {C}\mathscr {P}$-Condensing Operators on Semilattices, Join-Complete Lattices and Some Inverse Semigroups
Yong He , Wei Tian , Kar Ping Shum
Communications in Mathematics and Statistics ›› 2016, Vol. 4 ›› Issue (4) : 435 -447.
$\mathscr {P}$-Condense and $\mathscr {C}\mathscr {P}$-Condensing Operators on Semilattices, Join-Complete Lattices and Some Inverse Semigroups
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,
$\mathscr {P}$-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
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