Orders in Strict Regular Semigroups

 A subsemigroup S of a semigroup Q is an order in Q if for every there exist such that , where a and d are contained in (maximal) subgroups of Q, and and are their inverses in these subgroups. A regular semigroup S is strict if it is a subdirect product of completely (0-)simple semigroups. We construct all orders and involutions in Auinger’s model of a strict regular semigroup. This is used to find necessary and sufficient conditions on an involution on an order S in a strict regular semigroup Q for extendibility to an involution on Q.     

Bands of weakly r-Archimedean ordered semigroups

In this paper, some characterizations that an ordered semigroup S is a band of weakly r-archimedean ordered subsemigroups of S are given by some relations on S . We prove that an ordered semigroup S is a band of weakly r -archimedean ordered subsemigroups if and only if S is regular band of weakly r -archimedean ordered subsemigroups. Finally, we obtain that a negative ordered semigroup S is a band of weakly r-archimedean ordered subsemigroups of S if and only if S is a band of r-archimedean ordered subsemigroups of S . As an application the corresponding results on semigroups without order can be obtained by moderate modifications. August 27, 1999     

ON THE STRUCTURE OF REGULAR CRYPTO SEMIGROUPS

Superabundant semigroups are generalizations of completely regular semigroups written the class of abundant semigroups. It has been shown by Fountain that an abundant semigroup is superabundant if and only if it is a semilattice of completely J ^*-simple semigroups. Reilly and Petrich called a semigroup S cryptic if the Green's relation H is a congruence on S. In this paper, we call a superabundant semigroup S a regular crypto semigroup if H ^* is a congruence on S such that S/H ^* is a regular band. It will be proved that a superabundant semigroup S is a regular crypto semigroup if and only if S is a refined semilattice of completely J ^*-simple semigroups. Thus, regular crypto semigroups are generalization of the cryptic semigroups as well as abundant semigroups.     

*-orderable semigroups

Fix a *-orderable field k. We introduce the class of *-orderable semigroups as those semigroups with involution S for which the semigroup algebra kS endowed with the canonical involution admits a *-ordering. It is shown that this class is a quasivariety that is locally and residually closed. A cancellative nilpotent semigroup with involution is proved to be *-orderable if and only if it has unique extraction of roots. In general this equivalence fails, although every *-orderable semigroup has unique extraction of roots.     

Intuitionistic fuzzy semiprime ideals in ordered semigroups

We give characterizations of different classes of ordered semigroups by using intuitionistic fuzzy ideals. We prove that an ordered semigroup is regular if and only if every intuitionistic fuzzy left (respectively, right) ideal of S is idempotent. We also prove that an ordered semigroup S is intraregular if and only if every intuitionistic fuzzy two-sided ideal of S is idempotent. We give further characterizations of regular and intra-regular ordered semigroups in terms of intuitionistic fuzzy left (respectively, right) ideals. In conclusion of this paper we prove that an ordered semigroup S is left weakly regular if and only if every intuitionistic fuzzy left ideal of S is idempotent.     

Biordered Sets of Regular Semigroups with Inverse Transversals1

In a regular semigroup S, an inverse subsemigroup S° of S is called an inverse transversal of S if S° contains a unique inverse x° of each element x of S. An inverse transversal S° of S is called a Q-inverse transversal of S if S° is a quasi-ideal of S.If S is a regular semigroup with set of idempotents E then E is a biordered set. T.E. Hall obtained a fundamental regular semigroup T_E from the subsemigroup E which is generated by the set of idempotents of a regular semigroup. K.S.S. Nambooripad constructed a fundamental regular semigroup by a regular biordered set abstractly. In this paper, we discuss the properties of the biordered sets of regular semigroups with inverse transversals. This kind of regular biordered sets is called IT-biordered sets. We also describe the fundamental regular semigroup T_E when E is an IT-biordered set. In the sequel, we give the construction of an IT-biordered set by a left regular IT-biordered set and a right regular IT-biordered set.This project has been supported by the Provincial Natural Science Foundation of Guangdong Province, PR China     

Associate inverse subsemigroups of regular semigroups

By an associate inverse subsemigroup of a regular semigroup S we mean a subsemigroup T of S containing a least associate of each x∈S, in relation to the natural partial order ≤ _S . We describe the structure of a regular semigroup with an associate inverse subsemigroup, satisfying two natural conditions. As a particular application, we obtain the structure of regular semigroups with an associate subgroup with medial identity element. Research supported by the Portuguese Foundation for Science and Technology (FCT) through the research program POCTI.     

Cellularity of twisted semigroup algebras

In this paper, the cellularity of twisted semigroup algebras over an integral domain is investigated by introducing the concept of cellular twisted semigroup algebras of type JH. Partition algebras, Brauer algebras and Temperley-Lieb algebras all are examples of cellular twisted semigroup algebras of type JH. Our main result shows that the twisted semigroup algebra of a regular semigroup is cellular of type JH with respect to an involution on the twisted semigroup algebra if and only if the twisted group algebras of certain maximal subgroups are cellular algebras. Here we do not assume that the involution of the twisted semigroup algebra induces an involution of the semigroup itself. Moreover, for a twisted semigroup algebra, we do not require that the twisting decomposes essentially into a constant part and an invertible part, or takes values in the group of units in the ground ring. Note that trivially twisted semigroup algebras are the usual semigroup algebras. So, our results extend not only a recent result of East, but also some results of Wilcox.     

Variants of Regular Semigroups

Let S be a regular semigroup, and let a ∈ S . Then a variant of S with respect to a is a semigroup with underlying set S and multiplication \circ defined by x \circ y = xay . In this paper, we characterise the regularity preserving elements of regular semigroups; these are the elements a such that (S,\circ) is also regular. Hickey showed that the set of regularity preserving elements can function as a replacement for the unit group when S does not have an identity. As an application, we characterise the regularity preserving elements in certain Rees matrix semigroups. We also establish connections with work of Loganathan and Chandrasekaran, and with McAlister's work on inverse transversals in locally inverse semigroups. We also investigate the structure of arbitrary variants of regular semigroups concentrating on how the local structure of a semigroup affects the structure of its variants. May 24, 1999     

Abundant and ample straight left orders

A subsemigroup S of a semigroup Q is a straight left order in Q and Q is a semigroup of straight left quotients of S if every q ∈ Q can be written as for some with a b in Q and if, in addition, every element of S that is square cancellable lies in a subgroup of Q. Here a ^♯ denotes the group inverse of a in some (hence any) subgroup of Q. If S is a straight left order in Q, then Q is necessarily regular; the idea is that Q has a better understood structure than that of S. Necessary and sufficient conditions exist on a semigroup S for S to be a straight left order. The technique is to consider a pair of preorders on S. If such a pair satisfies conditions mimicking those satisfied by on a regular semigroup, and if certain subsemigroups of S are right reversible, then S is a straight left order. The conditions required for to satisfy are somewhat lengthy. In this paper we aim to circumvent some of these by specialising in two ways. First we consider only fully stratified left orders, that is, the case where (certainly the most natural choice for ) and the other is to insist that S be abundant, that is, every -class and every -class of S contains an idempotent. Our results may be used to show that the monoid of endomorphisms of a hereditary basis algebra of finite rank is a fully stratified straight left order. This revised version was published online in August 2006 with corrections to the Cover Date.     

A Structure Preserving Embedding of Semigroups Into Regular Semigroups

For any semigroup S a regular semigroup 𝒞(S) that embeds S can be constructed as the direct limit of a sequence of semigroups each of which contains a copy of its predecessor as a subsemigroup whose elements are regular. The construction is modified here to obtain an embedding of S into a regular semigroup R such that the nontrivial maximal subgroups of R are isomorphic to the Schützenberger groups of S and such that the restriction to S of any of Green's relations on R is the corresponding Green's relation on S.     

Some relations on completely regular semigroups

A semigroup S is called a Clifford semigroup if it is completely regular and inverse. In this paper, some relations related to the least Clifford semigroup congruences on completely regular semigroups are characterized. We give the relation between Y and ξ on completely regular semigroups and get that Y ^* is contained in the least Clifford congruence on completely regular semigroups generally. Further, we consider the relation Y ^*, Y, ν and ε on completely simple semigroups and completely regular semigroups. This work is supported by Leading Academic Discipline Project of Shanghai Normal University, Project Number: DZL803 and General Scientific Research Project of Shanghai Normal University, No. SK200707.     

Kakutani-type fixed point theorems: A survey

A Kakutani-type fixed point theorem refers to a theorem of the following kind: Given a group or semigroup S of continuous affine transformations s : Q → Q, where Q is a nonempty compact convex subset of a Hausdorff locally convex linear topological space, then under suitable conditions S has a common fixed point in Q, i.e., a point a ? Q{a \in Q} such that s(a) = a for each s ? S{s \in S}. In 1938, Kakutani gave two conditions under each of which a common fixed point of S in Q exists. They are (1) the condition that S be a commutative semigroup, and (2) the condition that S be an equicontinuous group. The present survey discusses subsequent generalizations of Kakutani’s two theorems above.     

Certain congruences on E-inversive E-semigroups

A semigroup S is called E-inversive if for every a ∈ S there exists x ∈ S such that ax is idempotent. S is called E-semigroup if the set of idempotents of S forms a subsemigroup. In this paper some special congruences on E-inversive E-semigroups are investigated, such as the least group congruence, a certain semilattice congruence, some regular congruences and a certain idempotent-separating congruence.     

Matrices of bisimple regular semigroups

A semigroup S is a matrix of subsemigroups S_iμ, i ε I, μ ε M if the S_iμ form a partition of S and S_iμS_jν≤S_iν for all i, j in I, μ, ν in M. If all the S_iμ are bisimple regular semigroups, then S is a bisimple regular semigroup. Properties of S are considered when the S_iμ are bisimple and regular; for example, if S is orthodox then each element of S has an inverse in every component S_iμ.     

Regular Semigroups Each of Whose Least Completely Simple Congruence Classes Has a Greatest Element

A regular semigroup S is termed locally F-regular, if each class of the least completely simple congruence ξ contains a greatest element with respect to the natural partial order. It is shown that each locally F-regular semigroup S admits an embedding into a semidirect product of a band by S/ξ. Further, if ξ satisfies the additional property that for each s ∈ S and each inverse (sξ)′ of sξ in S/ξ the set (sξ)′ ∩ V(s) is not empty, we represent S both as a Rees matrix semigroup over an F-regular semigroup as well as a certain subsemigroup of a restricted semidirect product of a band by S/ξ.     

On semigroups admitting ring structure

A right-chain semigroup is a semigroup whose right ideals are totally ordered by set inclusion. The main result of this paper says that if S is a right-chain semigroup admitting a ring structure, then either S is a null semigroup with two elements or sS=S for some s∈S. Using this we give an elementary proof of Oman’s characterization of semigroups admitting a ring structure whose subsemigroups (containing zero) form a chain. We also apply this result, along with two other results proved in this paper, to show that no nontrivial multiplicative bounded interval semigroup on the real line ℝ admits a ring structure, obtaining the main results of Kemprasit et al. (ScienceAsia 36: 85–88, 2010).