Lower Semimodular Inverse Semigroups, II

• Published in: Communications in algebra Vol. 39; no. 3; pp. 955 - 971
• Main Authors: Cheong, Kyeong Hee, Jones, Peter R.
• Format: Journal Article
• Language: English
• Published: Taylor & Francis Group 16.03.2011
• Subjects: Algebra; Decomposition; Divergence; Group theory; Inverse; Inverse semigroup; Lattice modules; Mathematical analysis; Semimodular; Subsemigroup lattice; Trees
• ISSN: 0092-7872; 1532-4125
• DOI: 10.1080/00927871003614439

The authors' description of the inverse semigroups S for which the lattice ℒℱ(S) of full inverse subsemigroups is lower semimodular is used to describe those for which (a) the lattice ℒ(S) of all inverse subsemigroups or (b) the lattice o(S) of convex inverse subsemigroups has that property. In each case, we show that this occurs if and only if the entire lattice is a subdirect product of ℒℱ(S) with ℒ(E S ), or o(E S ), respectively, where E S is the semilattice of idempotents of S; a simple necessary and sufficient condition is found for each decomposition. For a semilattice E, ℒ(E) is in fact always lower semimodular, and o(E) is lower semimodular if and only if E is a tree. The conjunction of these results leads to quite a divergence between the ultimate descriptions in the two cases, ℒ(S) and o(S), with the latter being substantially richer.