Absolutely flat completely 0-simple semigroups of left quotients
It is proposed a semigroup analogue of Goldie's theorem describing rings which have a simple Artinian ring of left quotients. In this paper the author is concerned with a semigroup analogue of the classical ring of quotients, first proposed by \textit{J. B. Fountain} and \textit{M. Petrich} [J. Algebra 101, 365-402 (1986; Zbl 0589.20041)]. If Q is a semigroup of left quotients of S then S is called a left order in Q. If Q is both a semigroup of left quotients of S and a semigroup of right quotients of S then Q is called a semigroup of quotients and S is called an order in Q. Fountain and Petrich described all orders in completely 0-simple semigroups. If one wishes to classify the left orders in a given class of semigroups, then one requires the semigroups of left quotients from that class to be unique up to semigroups. Unfortunately this is not true for completely 0-simple semigroups. The notion of absolutely flat semigroups one can find in [\textit{S. Bulman-Fleming} and \textit{K. McDowell}, Pac. Math. 107, 319-333 (1983; Zbl 0488.20054)]. The author proves the uniqueness of absolutely flat completely 0-simple semigroups of left (right) quotients. Further it is given a set of necessary and sufficient conditions for a semigroup S to be a left order in an absolutely flat completely 0-simple semigroup. In conclusion these results are applied to Brandt semigroups and abundant semigroups.
orders in completely 0-simple semigroups
absolutely flat completely 0-simple semigroups
scientific article; zbMATH DE number 4075400
scientific article