The Lattice of Full Subsemigroups of an Inverse Semigroup

Authors: Tian, Zhenji¹; Xu, Zongben²

Source: Semigroup Forum, Volume 73, Number 3, December 2006 , pp. 457-469(13)

Publisher: Springer

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Abstract:

In this paper, we consider the lattice Subf S of full subsemigroups of an inverse semigroup S. Our first main theorem states that for any inverse semigroup S, Subf S is a subdirect product of the lattices of full subsemigroups of its principal factors, so that Subf S is distributive [meet semidistributive, join semidistributive, modular, semimodular] if and only if the lattice of full subsemigroups of each principal factor is. To examine such inverse semigroups, therefore, we need essentially only consider those which are 0-simple. For a 0-simple inverse semigroup S (not a group with zero), we show that in fact each of modularity, meet semidistributivity and join semidistributivity of Subf S is equivalent to distributivity of S, that is, S is the combinatorial Brandt semigroup with exactly two nonzero idempotents and two nonidempotents. About semimodularity, however, we concentrate only on the completely 0-simple case, that is, Brandt semigroups. For a Brandt semigroup S (not a group with zero), semimodularity of Subf S is equivalent to distributivity of Subf S. Finally, we characterize an inverse semigroup S for which Subf S is a chain.

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s00233-006-0626-1

Affiliations: 1: Email: zjtian@lut.cn 2: Email: zbxu@mail.xjtu.edu.cn

Publication date: 2006-12-01