Tight representations of semilattices and inverse semigroups

Details Tight representations of semilattices and inverse semigroups Tight representations of semilattices and inverse semigroups

2007-03-14

arxiv.org/abs/math/0703401v1

Semigroup Forum, 79 (2009), 159

Mathematics

Representation Theory

24 pages, 12 point type, no figures. This is a paper on representations of inverse semigroups

Scientific paper

By a "Boolean inverse semigroup" we mean an inverse semigroup whose semilattice of idempotents is a Boolean algebra. We study representations of a given inverse semigroup S in a Boolean inverse semigroup which are "tight" in a certain well defined technical sense. These representations are supposed to preserve as much as possible any trace of "Booleannes" present in the semilattice of idempotents of S. After observing that the Vagner-Preston representation is not tight, we exhibit a canonical tight representation for any inverse semigroup with zero, called the "regular representation". We then tackle the question as to whether this representation is faithful, but it turns out that the answer is often negative. The lack of faithfulness is however completely understood as long as we restrict to "continuous" inverse semigroups, a class generalizing the E*-unitaries.

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