The Lattice of Full Subsemigroups of an Inverse Semigroup |
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Authors: | Zhenji Tian Zongben Xu |
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Institution: | (1) School of Science, Lanzhou University of Technology, Lanzhou 730050, P.R. China;(2) School of Science, Xi'an Jiaotong University, Xi'an 710049, P.R. China |
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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. |
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Keywords: | |
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