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Glrac semigroups are common generalizations of left GC-lpp semigroups and left inverse semigroups. And, such a semigroup is a left restriction semigroup if and only if the projection set is a semilattice. So, glrac semigroup is also a generalization of left restriction semigroup. Permissible subsets of a glrac semigroup are introduced in this paper. In terms of permissible subsets, we define (uniquely) factorizable glrac semigroups and (uniquely) almost factorizable glrac semigroups. Many characterizations of (uniquely) factorizable glrac semigroups and (uniquely) almost factorizable glrac semigroups are obtained. As their applications, we establish the structures of uniquely factorizable left GC-lpp semigroups (left inverse semigroups, inverse semigroups, ample semigroups, left restriction semigroup, restriction semigroups) and uniquely almost factorizable left GC-lpp semigroups (left inverse semigroups, inverse semigroups, ample semigroups, left restriction semigroup, restriction semigroups). Our results enrich and extend the related results of almost factorizable restriction semigroups.

Abundant semigroups originate from p.p. rings and are generalizations of regular semigroups. The main aim of this paper is to study the primeness and the primitivity of abundant semigroup algebras. We introduce and studyD∗{{\mathcal{D}}}^{\ast }-graphs and Fountain matrices of a semigroup. Based onD∗{{\mathcal{D}}}^{\ast }-graphs and Fountain matrices, we determine when a contracted semigroup algebra of a primitive abundant semigroup is prime (respectively, semiprime, semiprimitive, or primitive). It is well known that for any algebraA{\mathcal{A}}with unity,A{\mathcal{A}}is primitive (prime) if and only if so isMn(A){M}_{n}\left({\mathcal{A}}). Our results can be viewed as some kind of generalizations of such a known result. In addition, it is proved that any contracted semigroup algebra of a locally ample semigroup whose set of idempotents is locally finite (respectively, locally pseudofinite and satisfying the regularity condition) is isomorphic to some contracted semigroup algebra of primitive abundant semigroups. Moreover, we obtain sufficient and necessary conditions for these classes of contracted semigroup algebras to be prime (respectively, semiprime, semiprimitive, or primitive). Finally, the structure of simple contracted semigroup algebras of idempotent-connected abundant semigroups is established. Our results enrich and extend the related results on regular semigroup algebras.

It is proved that any Azumaya semigroup algebra of an abundant semigroup over a field is a subdirect product of matrix algebras over Azumaya semigroup algebras of cancellative monoids. In particular, it is proved that for an abundant semigroup S with finitely many idempotents and a field K, $$K_0[S]$$ is Azumaya if and only if $$K_0[S]$$ is isomorphic to the direct sum of finite matrix algebras of Azumaya algebras of cancellative submonoids. This gives an affirmative answer to an open problem of Okninski in his monograph (Semigroup Algebra, Marcel Dekker, New York, 1991) for the case that the semigroup is a finite abundant semigroup.

It is proved that for an IC abundant semigroup (a primitive abundant semigroup; a primitively semisimple semigroup) S and a field K, if K 0[S] is right (left) self-injective, then S is a finite regular semigroup. This extends and enriches the related results of Okniński on self-injective algebras of regular semigroups, and affirmatively answers Okniński’s problem: does that a semigroup algebra K[S] is a right (respectively, left) self-injective imply that S is finite? (Semigroup Algebras, Marcel Dekker, 1990), for IC abundant semigroups (primitively semisimple semigroups; primitive abundant semigroups). Moreover, we determine the structure of K 0[S] being right (left) self-injective when K 0[S] has a unity. As their applications, we determine some sufficient and necessary conditions for the algebra of an IC abundant semigroup (a primitively semisimple semigroup; a primitive abundant semigroup) over a field to be semisimple.

Strict RA semigroups are common generalizations of ample semigroups and inverse semigroups. The aim of this paper is to study algebras of strict RA semigroups. It is proved that any algebra of strict RA semigroups with finite idempotents has a generalized matrix representation whose degree is equal to the number of non-zero regular 𝓓-classes. In particular, it is proved that any algebra of finite right ample semigroups has a generalized upper triangular matrix representation whose degree is equal to the number of non-zero regular 𝓓-classes. As its application, we determine when an algebra of strict RA semigroups (right ample monoids) is semiprimitive. Moreover, we prove that an algebra of strict RA semigroups (right ample monoids) is left self-injective iff it is right self-injective, iff it is Frobenius, and iff the semigroup is a finite inverse semigroup.

The notion of *-normal idempotents is introduced. The structure theorem for abundant semigroups with a *-normal idempotent is obtained. As its applications, we establish the construction theorem of naturally ordered abundant semigroups with a greatest idempotent.

Good congruences are defined as congruences preserving $$\mathcal L^*$$ -classes and $$\mathcal R^*$$ -classes. In this paper the good congruences on abundant semigroups generated by Green’s $$*$$ -relations are investigated. The related results for regular semigroups are enriched and extended.

A semigroup is called almost idempotent-free if it has exactly one nonzero \({\mathcal {J}}\)-class containing idempotents. It is mainly investigated when almost idempotent-free semigroup algebras are Azumaya algebras. Necessary and sufficient conditions for semigroup algebras of an almost idempotent-free semigroup with certain properties to be Azumaya are obtained.

We prove that the semigroup algebra of an ample semigroup \(S\) over a field is Frobenius if and only if \(S\) is a finite inverse semigroup.

In this paper, we establish a structure theorem of naturally ordered rpp semigroups with max-idempotents. Results on naturally ordered semigroups with max-idempotents in the literature are extended and amplified. Some previous results obtained by Blyth, McFadden, Guo and Xie on regular and abundant ordered semigroups are extended to ordered rpp semigroups.

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.

An adequate semigroup S is called ample if ea = a(ea)* and ae = (ae)†a for all a ∈ S and e ∈ E(S). Inverse semigroups are exactly those ample semigroups that are regular. After obtaining some characterizations of finite ample semigroups, it is proved that semigroup algebras of finite ample semigroups have generalized triangular matrix representations. As applications, the structure of the radicals of semigroup algebras of finite ample semigroups is obtained. In particular, it is determined when semigroup algebras of finite ample semigroup are semiprimitive.

Abundant semigroups expressed as a perfect rectangular band of adequate semigroups will be investigated. A known theorem of Pastijn–Petrich in 1984 on perfect rectangular bands of inverse semigroups will be generalized from the class of regular semigroups to the class of abundant semigroups. A Rees matrix representation of such abundant semigroups will be established and some characterization theorems will be consequently given.

We study a class of special strongly rpp semigroups, namely, the class of super rpp semigroups. These super rpp semigroups are generalizations of both superabundant semigroups and Clifford semigroups within the class of rpp semigroups. In particular, we prove that a super rpp semigroup is a semilattice of D (l)-simple strongly rpp semigroups. Our result not only generalizes a well-known theorem of Clifford in the class of completely regular semigroups but also strengthens some structure theorems obtained by Ren-Shum for superabundant semigroups which are orthodox. Some special super rpp semigroups are considered and discussed.

In this paper, we study some special residuated lattices, namely, conical idempotent residuated lattices. After obtaining some properties of such residuated lattices, we establish a structure theorem for conical idempotent residuated lattices.

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 concept of wreath product of semigroups was first introduced by Neumann in 1960, and later on, this concept was used by Preston to investigate the structure of some inverse semigroups. In this paper, we modify the wreath product given by Neumann and Preston to study the structure of some generalized Clifford semigroups. In particular, we prove that a semigroup is a left C-rpp semigroup if and only if it is the wreath product of a left regular band and a C-rpp semigroup. Our result provides a new insight to the structure of left C-rpp semigroups.

The concept of wreath product of semigroups was initiated by Neumann in 1960, and later on, his concept was used by Preston to investigate the structure of some inverse semigroups. Recently, we start to investigate the structure of left C-rpp semigroups by using wreath products. In this paper, we modify the wreath product to “dual wreath product” so that we can study the structure of right C-rpp semigroups. We prove that a semigroup is a right C-rpp semigroup if and only if it is the dual wreath product of a right regular band and a C-rpp semigroup. Our theorem provides new insight to the structure of right C-rpp semigroups. In particular, a recent result given by Ren and Shum for right C-rpp semigroups is strengthened and enriched.

As an analogue of right C-rpp semigroups and a dual of left C-wrpp semigroups, right C-qrpp semigroups were defined and investigated by Rao-Guo [10]. In this paper, we continue the research of such semigroups and obtain a structure of right C-qrpp semigroups. It is proved that a semigroup is a right C-qrpp semigroup if and only if it is isomorphic to a right Δ-product of a C-wrpp semigroup and a right regular band.

An l.p.p. ring satisfying the primitive idempotent condition is called left *-semisimple if it also satisfies the ascending condition on right *ideals. We prove that a left *-semisimple ring can be uniquely decomposed into a direct product of a finite number of full matrix rings over certain domains, up to isomorphism. In fact, such left *-semisimple rings are generalized semisimple rings and, in particular, they are semisimple when they are finite. Mathematics Subject Classification: 16W60

It is well known that the rpp semigroups are generalized regular semigroups. In order to further investigate the structure of rpp semigroups, we introduce the type-(I, L ∗ ) factorizable monoids. In this paper, we study such factorizable rpp monoids admitting a left univocal type-(I, L ∗ ) factorization. In particular, we prove that a semigroup S is a factorizable rpp monoid admitting a left univocal type-(I, L ∗ ) factorization if and only if S is isomorphic to a semidirect product of a band and a left cancellative monoid. Some results of Catino and Tolo on factorizable semigroups are extended and amplified.

The investigation of regular F-abundant semigroups is initiated. In fact, F-abundant semigroups are generalizations of regular cryptogroups in the class of abundant semigroups. After obtaining some properties of such semigroups, the construction theorem of the class of regular F-abundant semigroups is obtained. In addition, we also prove that a regular F-abundant semigroup is embeddable into a semidirect product of a regular band by a cancellative monoid. Our result is an analogue of that of Gomes and Gould on weakly ample semigroups, and also extends an earlier result of O'Carroll on F-inverse semigroups.

The aim of this paper is to study the congruence extension property and the ideal extension property for compact semigroups. We present a characterization of compact semigroups with the ideal extension property and prove that each compact semigroup with the congruence extension property also has the ideal extension property.

In this paper, the translational hull of a type-A semigroup is considered. Our main result is to prove that the translational hull of a semilattice of cancellative monoids (respectively a proper type-A semigroup, an E-reflexive type-A semigroup) is still the same type of semigroups. This answers an open problem posted by Petrich on translational hulls of semigroups in 1984.

The aim of this paper is to study semigroups with the ideal extension property. We establish a structure theorem for semigroups with the ideal extension property.

The aim of this paper is to study a class of left abundant semigroups, so-called abundant left C-lpp proper semigroups including left type A proper semigroups and right inverse proper semigroups as its subclasses. A structure theorem similar to McAlister’s for inverse proper semigroups is obtained. As its application, it is verified that any abundant left C-lpp proper semigroup can be embedded into a semidirect product of a left regular band by a cancellative monoid.

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.

Denote the2×2upper triangular matrix rings overℤandℤpbyUTM2(ℤ)andUTM2(ℤp), respectively. We prove that if a ringRis a p.p.-ring, thenRis reduced if and only ifRdoes not contain any subrings isomorphic toUTM2(ℤ)orUTM2(ℤp). Other conditions for a p.p.-ring to be reduced are also given. Our results strengthen and extend the results of Fraser and Nicholson on r.p.p.-rings.

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.

An adequate semigroup S is said to be ample if for any e2 = e, a ∈ S, ae = (ae)†a and ea = a(ea)*. It is well known that inverse semigroups are ample semigroups. The purpose of this paper is to study matrix representations of an ample semigroup. Some properties of ample semigroups are obtained. It is proved that any indecomposable good matrix representations of an ample semigroup can be constructed by using those of weak Brandt semigroups.

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.

The congruence extension property (CEP) of semigroups has been extensively studied by a number of authors. We call a compact semigroup S an Ω -compact semigroup if the set of all regular elements of S forms an ideal of S . In this note, we characterize the Ω -compact semigroup having (CEP). Our result extends a recent result obtained by X.J. Guo on the congruence extension property of strong Ω -compact semigroups which is a semigroup containing precisely one regular D -class.