A Class of Irreducible matrix representations of an Arbitrary Inverse Semigroup

By a ‘representation’ we shall mean throughout a representation by n × n matrices with entries from an arbitrary field. Elsewhere [9] the author has introduced the concept of a principal representation of a semigroup S (see § 3 below for the definition) and has shown that if S satisfies the minimal condition on principal ideals then every irreducible representation is of this type. Moreover, if S satisfies the minimal conditions on both principal left and right ideals, which together imply the minimal condition on principal two-sided ideals [6, Theorem 4], the irreducible representations of S can ultimately be expressed explicitly in terms of group representations.