Closed inverse subsemigroup lattices of inverse semigroups

Kyeong Hyeui Cheong, Marquette University

Abstract

The final goal of the dissertation is to find certain criteria in order for the lattices of closed, E-closed or convex inverse subsemigroups of inverse semigroups to be distributive, modular, semimodular, complemented or Boolean, etc. For the lattices ${\cal LC}$(S) of closed inverse subsemigroups of inverse semigroups S with the semilattices E of idempotents, we look for the conditions for the map: A $\mapsto$ (A $\cap$ E,E $\uparrow$ VA), A $\in$ ${\cal LC}$(S), where E $\uparrow$ the closure of E and V the join, to be a subdirect product representation, in which case the properties in question of ${\cal LC}$(S) are determined by those of both ${\cal LC}$(E) and ($E\uparrow)$. The same approach applies analogously to other types of lattices of concern. For the case of ${\cal LC}$(S), the ideas are developed to characterize inverse semigroups S with ${\cal LC}$(S) Boolean by: The semilattice E is a locally finite dually generalized Boolean lattice; Every element of S has period a product of distinct primes; The union of groups is a E-unitary direct product of the semilattice and the group of units. Inverse semigroups whose E-closed inverse subsemigroup lattices are distributive are determined by: If i $\ge$ $aea\sp{-1}$ in S and $i,e\in E$, then there exist $h\in E\cap < a > \uparrow$ and $k \in E \cap e\uparrow$ such that i = hk; If x $\ge$ y and y $\not\in\ E$, then y is in the convex inverse subsemigroup generated by x; The lattice of full inverse subsemigroups is distributive. The characterization of inverse semigroups whose convex inverse subsemigroup lattices are lower semimodular is given by: The semilattice E of idempotents forms a tree; If x $\ge$ y and y $\not\in$ E, then y is in the convex inverse subsemigroup generated by x; The lattice of full inverse subsemigroups is lower semimodular.

Recommended Citation

Kyeong Hyeui Cheong, "Closed inverse subsemigroup lattices of inverse semigroups" (January 1, 1994). Dissertations (1962 - 2010) Access via Proquest Digital Dissertations. Paper AAI9505907.
http://epublications.marquette.edu/dissertations/AAI9505907

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