# Closed inverse subsemigroup lattices of inverse semigroups

#### 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

Cheong, Kyeong Hyeui, "Closed inverse subsemigroup lattices of inverse semigroups" (1994). *Dissertations (1962 - 2010) Access via Proquest Digital Dissertations*. AAI9505907.

https://epublications.marquette.edu/dissertations/AAI9505907