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Uniform representation of semigroups is introduced. It is proved that any uniform representation of an ample semigroup can be expressed as the direct sum of some representations obtained via homogenous representations on primitive adequate semigroups. Also, we give the structure of homogenous representations of primitive adequate semigroups. In addition, we consider indecomposable uniform representations of ample semigroups and their constructions.

Rees matrix covers for regular semigroups were first studied by McAlister in 1984. Lawson extended McAlister's results to abundant semigroups in 1987. We consider here a semigroup whose set of regular elements forms a subsemigroup, named tight semigroups. In this paper, it is proved that an abundant semigroup is tight and locally E-solid if and only if it is an F-local isomorphic image of an abundant Rees matrix semigroup [Formula: see text] over a tight E-solid abundant semigroup T, where the entries of the sandwich matrix P of [Formula: see text] are regular elements of T. Our results enrich the result of Lawson on Rees matrix covers for a class of abundant semigroups and extend the results of McAlister on Rees matrix covers for regular semigroups.

We generalize the F-inverse semigroups within the class of lpp-semigroups by using McAlister's approach and FGC-systems. Consider a left GC - lpp monoid M. If M is lpp, then M is called a left F-pseudo group and for brevity, we call the semi-direct product of a left regular band and a cancellative monoid a twisted left cryptic group. In this paper, the structures of left F-pseudo groups are investigated. It is shown that a left F-pseudo group whose minimum right cancellative monoid congruence is cancellative can be embedded into a twisted left cryptic group. This result generalizes a number of known results in F-inverse semigroups previously given by C. C. Edwards, R. B. McFadden, L. O'Carrol, X. J. Guo and others. In particular, a new method constructing F -right inverse semigroups is provided.

A semigroup S is called rpp if all right principal ideals of S, regarded as S1-systems, are projective. An rpp semigroup S is said to be strongly rpp if for any a ∈ S, there exists a unique idempotent e such that [Formula: see text] and a = ea. In this paper, we show that a [Formula: see text]-simple strongly rpp semigroup can be expressed by a Rees matrix semigroup over a left cancellative monoid and conversely. Our result generalizes the classical theorem of Rees in 1940 and also amplifies the Rees theorem in semigroup given by Lallement and Petrich in 1969.

The properties of right ample semigroups have been extensively considered and studied by many authors. In this paper, we concentrate on the matrix representations of right ample semigroups. The (left; right) uniform matrix representation is initially defined. After some properties of left uniform matrix representations of a right ample semigroup are given, we prove that any irreducible left uniform representations of a right ample semigroup can be obtained by using an irreducible left uniform representation of some primitive right ample semigroup. In particular, a construction theorem of prime left uniform representation of right ample semigroups is established.

An idempotent-connected abundant semigroup S is a locally ample semigroup if for any idempotent e of S, the local submonoid eSe of S is an ample subsemigroup of S. Clearly, an ample semigroup is a locally ample semigroup. In this paper, it is proved that the semigroup algebra of a finite locally ample semigroup is isomorphic to the semigroup algebra of an associate primitive abundant semigroup. As an application of this result, we characterize Jacobson radicals of finite locally ample semigroup algebras.

In this paper, we prove that any prime irreducible representation of a left ample semigroup being eventually regular can be constructed by some irreducible representation of some groups. This result enriches and extends the related results of W. D. Munn on prime irreducible representations of an inverse semigroup.