Clarke and Monzo defined in [3] a construction called a generalized inflation of a semigroup. It is always the case that any inflation of a semigroup is a generalized inflation, and any generalized inflation of a semigroup is a null extension of the semigroup. Clarke and Monzo proved that any associative null extension of a base semigroup which is a union of groups is in fact a generalized inflation. In this paper we study null extensions and generalized inflations of Brandt semigroups. We first prove that any generalized inflation of a Brandt semigroup is actually an inflation of the semigroup. This answers a question posed by Clarke and Monzo in [3]. Then we characterize associative null extensions of Brandt semigroups, and show that there are associative null extensions of Brandt semigroups which are not generalized inflations.

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doi.org/10.1007/s00233-006-0663-9
Semigroup Forum
School of Mathematics and Statistics

Wang, Q, & Wismath, S.L. (2007). Null extensions and generalized inflations of brandt semigroups. Semigroup Forum, 74(2), 274–292. doi:10.1007/s00233-006-0663-9