Tight Representations of 0-𝐸-Unitary Inverse Semigroups

Abstract

We study the tight representation of a semilattice in {0,1} by some examples. Then we introduce the concept of the complex tight representation of an inverse semigroup 𝑆 by the concept of the tight representation of the semilattice of idempotents 𝐸 of 𝑆 in {0,1}. Specifically we describe the tight representation of a 0-𝐸-unitary inverse semigroup and prove that if 𝜎 is a tight semilattice representation of the 0-𝐸-unitary inverse semigroup 𝑆 in {0,1}, then 𝜎 is a complex tight representation.