We introduce the class of network right *-abundant semigroups. These are based on networks that extend the notion of a directed graph. This class properly contains the class of graph inverse semigroups. We investigate the structure of network right *-abundant semigroups. We show that two network right *-abundant semigroups are isomorphic if and only if the underlying networks are isomorphic.

The equivalences L-*,L-similar to and R-*,R-similar to on a semigroup S provide natural generalizations of Green's relations G and R, respectively. A semigroup S is said to be an m-wide semigroup if each L-*,L-similar to -class and each R-*,R-similar to of S contains an idempotent. We consider some special m-wide semigroups, namely, the generalized left inverse semigroups. After obtaining some properties and characterizations of such a semigroup, we establish a structure theorem of this class of semigroups. This extends the results on left inverse semigroups given by M. Yamada and on L*-inverse semigroups obtained by Ren-Shum to r-wide semigroups.

Quasi-automatic semigroups are extensions of a Cayley graph of an automatic group. Of course, a quasi-automatic semigroup generalizes an automatic semigroup. We observe that a semigroup [Formula: see text] may be automatic only when [Formula: see text] is finitely generated, while a semigroup may be quasi-automatic but it is not necessary finitely generated. Similar to the usual automatic semigroups, a quasi-automatic semigroup is closed under direct and free products. Furthermore, a semigroup [Formula: see text] is graph automatic if and only if [Formula: see text] with a zero element adjoined is graph automatic, and also a semigroup [Formula: see text] is graph automatic if and only if [Formula: see text] with an identity element adjoined is graph automatic. However, the class of quasi-automatic semigroups is a much wider class than the class of automatic semigroups. In this paper, we show that every automatic semigroup is quasi-automatic but the converse statement is not true (see Example ??). In addition, we notice that the quasi-automatic semigroups are invariant under the changing of generators, while a semigroup may be automatic with respect to a finite generating set but not the other. Finally, the connection between the quasi-automaticity of two semigroups [Formula: see text] and [Formula: see text], where [Formula: see text] is a subsemigroup with finite Rees index in [Formula: see text] will be investigated and considered.

A kind of weakly rpp semigroups,namely,weakly left C-rpp semigroups,is studied.After giving a semilattice decompositions of weakly left C-rpp semigroups,we prove that every weakly left C-rpp semigroup can be expressed as a left cross product of a strong semilattice of unipotent monoids and a left regular band.These work may be regarded as extending the results in[Semigroup Forum,1976/77,13(3):229-237]for C-rpp semigroups and in[Semigroup Forum,1995,50(1):9-23]for left C-rpp semigroups.

The theory of congruences is an important topic in semigroup theory. There are plenty of results about congruence theory on regular semigroups. Abundant semigroups are generalizations of regular semigroups. There have been many attempts to work on congruence theory on abundant semigroups. In this paper, some results about congruences theory on abundant semigroups are summarized. We focus our survey mainly on the work of the Chinese people, of course also related to some of the work overseas.

The purpose of this paper is to investigate restriction ω -semigroups. Here a restriction ω -semigroup is a generalisation of an inverse ω -semigroup. We give a description of a class of restriction ω -semigroups, namely, restriction ω -semigroups with an inverse skeleton. We show that a restriction ω -semigroup with an inverse skeleton is an ideal extension of a (cid:2) J -simple restriction ω -semigroup by a restriction semigroup with a finite chain of projections with a zero adjoined. This result is analogous to Munn’s result for inverse ω -semigroups. In addition, we show that the Bruck–Reilly semigroup of a strong semilattice of monoids indexed by a finite chain is a (cid:2) J -simple restriction ω -semigroup with an inverse skeleton, conversely, every (cid:2) J -simple restriction ω -semigroup with an inverse skeleton arises in this way.

A semigroup S is called a weakly abundant semigroup if its every (L) over tilde -class and every (R) over tilde -class contains an idempotent. Our purpose is to study an analogue of orthodox semigroups in the class of weakly abundant semigroups. Such an analogue is called a left quasi-abundant semigroup, which is a weakly abundant semigroup with a left quasi-normal band of idempotents and having the congruence condition (C). To build our main structure theorem for left quasi-abundant semigroups, we first give a sufficient and necessary condition of the idempotent set E(S) of a weakly abundant semigroup S being a left quasi-normal band. And then we construct a left quasi-abundant semigroup in terms of weak spined products. Such a result is a generalisation of that of Guo and Shum for left semi-perfect abundant semigroups. In addition, we consider a type Q semigroup which is a left quasi-abundant semigroup having the PC condition.

研究了具左中心投射元的U-rpp半群(简称左U-rpp半群).这类半群是具左中心幂等元的富足半群在U-rpp半群类中的一个自然推广.在定义具左中心投射元的U-rpp半群和U-左可消半群之后,借助具左中心投射元的U-rpp半群上的半格同余,建立了此类半群的一个代数结构.证明了一个半群(S,U)是具左中心投射元的U-rpp半群,当且仅当(S,U)是U左可消幺半群和右零带的直积的半格;当且仅当(S,U)是U-左可消幺半群和右零带的直积的强半格.

We introduce the relations [Formula: see text] and [Formula: see text] with respect to a subset U of idempotents. Based on [Formula: see text] and [Formula: see text], we define a new class of semigroups which we name U-concordant semigroups. Our purpose is to describe U-concordant semigroups by generalized categories over a regular biordered set. We show that the category of U-concordant semigroups and admissible morphisms is isomorphic to the category of RBS generalized categories and pseudo functors. Our approach is inspired from Armstrong’s work on the connection between regular biordered sets and concordant semigroups. The significant difference in strategy is by using RBS generalized categories equipped with pre-orders, we have no need to discuss the quotient of a category factored by a congruence.

利用(?,～)-好同余对刻画了完全J?,～-单半群上的(?,～)-好同余.此结果将正则半群中有关完全单半群上同余的相关结论推广到r-wide半群中,为下一步研究超r-wide半群上的好同余奠定了基础.

A class of weakly U-abundant semigroups satisfying the congruence condition (C) is investigated, which contains both the class of regular semigroups and the class of abundant semigroups as its subclasses. In particular, the class of weakly U-abundant semigroups satisfying the congruence condition (C) with Ehresmann transversals is studied. After giving some properties of U-abundant semigroups with Ehresmann transversals, we establish some characterization theorems for such semigroups. Our results may be regarded as an extension of the work of Blyth, McAlister and McFadden in regular semigroups and that of El-Qallali, Guo and Chen in abundant semigroups.

As a generalization of left inverse semigroups in the class of regular semi- groups, we consider a class of semigroups which we name E -inverse semi groups. After introducing the notion of left circle product for semigroups, we give a construction method of such a semigroups. It is proved that a semigroup S is an f -inverse semigroup if and only if S can be expressed as a left circle product of an E -ample semigroup and a left regular band. Our work may, be regarded as extending the result of Yamada for left inverse semigroups and :he structure theorem obtained by Ren-Shum for f* -inverse semigroups.

As a generalization of left inverse semigroups in the class of regular semigroups, we consider a class of semigroups which we name ℒ̃-inverse semigroups. After introducing the notion of left circle product for semigroups, we give a construction method of such a semigroups. It is proved that a semigroup S is an ℒ̃-inverse semigroup if and only if S can be expressed as a left circle product of an E-ample semigroup and a left regular band. Our work may be regarded as extending the result of Yamada for left inverse semigroups and the structure theorem obtained by Ren-Shum for ℒ*-inverse semigroups.

We use Malcev product of semigroups satisfying some axiomatic conditions to describe the structure of superabundant semigroups and some of its subclasses. Some characterization theorems of these kinds of semigroups are given.

Let S be a nil-extension of a Clifford semigroup K by a nil semigroup Q =S/K.By introducing a concept of admissible congruence pairs (δ,ω),where δis a congruence on a nil semigroup Q and ωis a congruence on a Clifford semigroup K respectively,it is proved that every congruence σon S can be uniquely represented by an admissible con-gruence pair on S.In addition,for any congruence σon S,suppose that σK is a restriction of σon a Clifford semigroup K,that is,σK =σ|K and σQ =(σ∨ρK)/ρK,where ρK is a Rees congruence on S induced by a ideal K of S,it is proved that there is an order-preserving bijection Γ:σ→(σQ,σk)from the set of all congruences on S onto the set of all admissi-ble congruence pairs on S.Finally,a condition has been given for a congruence which is a regular congruence on S.

By using malcev product of semigroups and axiomatic conditions, we describe superabundant semi-groups and its several subclasses. Some characterizations of these semigroups are given in this paper.

定义完美l-ample半群,并研究具有左中心幂等元的完美l-ample半群的半格分解.利用半格分解,证明了半群S为具有左中心幂等元的完美l-ample半群,当且仅当S为直积Mα×∧α的强半格,其中Mα是右可消幂幺半群,∧α是右零带.这一结果为具有左中心幂等元的完美l-ample半群结构的建立奠定了基础.

In this paper, we study the semilattice decomposition of U-abundant semigroups with left central idempotents. By using this semilattice decomposition, it is proved that a semigroup S is a U-abundant semigroup with left central idempotents if and only if it is a strong semilattice of a direct product Mα×Λα, where Mα is a unipotent monoid and Λα is a right zero band. This result is the basis of the establishing of the structure theorem of U-abundant semigroups with left central idempotents.

In this paper, (∗,∼)-good congruences on normal ortho-lc-monoids are characterized by means of (∗,∼)-good congruence pair. The result about congruences on normal orthogroups in regular semigroups is generalized to r-wide semigroups.