On Orders in Normal Cryptogroups

Abstract

A semigroup S is an order in a semigroup Q if every element of Q is of the form q = a −1 b = cd −1 for some a, b, c, d ∈ S, where a −1 and d −1 are inverses within group H-classes of Q. We study orders in normal cryptogroups based on considering the restrictions of Green's relations of Q to S. This produces certain relations which help us to gain an overview of all normal cryptogroups Q in which S is an order. Distinguished orders play an important role in this context. Mathematics Subject Classification (2010). 20M10.