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The category of representations of a completely 0-simple semigroup

A. H. Clifford [1], [2] has shown that all finite dimensional representations of a completely 0-simple semigroup S over a field Φ.phi; can be obtained as extensions of those of its maximal subgroups and has given a method for constructing all such representations. This representation theory depends strongly on the fact the representations under consideration are finite dimensional and is not adequate to deal with the infinite dimensional case or with representations over arbitrary rings. In order to determine the structure of the (contracted) algebra Φ(S) of S modulo its radical, one has to consider representations which are not finite dimensional or over fields; c.f. ‘6’. Hence Clifford's theory does not suffice for this purpose.