Regularity of semigroup rings

Sufficient conditions are obtained in order that the semigroup rings of some semigroups be regular. For some inverse, orthodox and completely simple semigroups, the necessary and sufficient conditions for the regularity of the semigroup rings are found. The author wishes to thank Prof. Y. Q. Guo for his best help and advice. Key Words and Phrases: (von Neumann) regularity, semigroup ring, inverse semigroup, orthodox semigroup, completely [0-] simple semigroup.     

Representations of involutive semigroups

Involutive [topological] semigroups are semigroups endowed with an involutive antiautomorphism. A representation of such a semigroup means an involution preserving [weakly continuous] morphism into the algebra of bounded operators on a Hilbert space. We develop a representation theory of involutive [topological] semigroups based on positive definite functions on them. We do not generally assume the existence of an identity element. This makes the proofs of some results harder, but most results hold in this general setting. The author thanks the referee for many constructive suggestions to improve the exposition of the paper.     

Derived Rees Matrix Semigroups as Semigroups of Transformations

An ordered pair (e,f) of idempotents of a regular semigroup is called a skew pair if ef is not idempotent whereas fe is idempotent. Previously [1] we have established that there are four distinct types of skew pairs of idempotents. We have also described (as quotient semigroups of certain regular Rees matrix semigroups [2]) the structure of the smallest regular semigroups that contain precisely one skew pair of each of the four types, there being to within isomorphism ten such semigroups. These we call the derived Rees matrix semigroups. In the particular case of full transformation semigroups we proved in [3] that T_X contains all four skew pairs of idempotents if and only if |X| ≥ 6. Here we prove that T_X contains all ten derived Rees matrix semigroups if and only if |X| ≥ 7.     

On the weak regularity of semigroup rings

Sufficient conditions are obtained under which the semigroup ring of a semigroup, in particular, of an inverse semigroup, is weakly regular. For some inverse semigroups and some orthodox semigroups, the necessary and sufficient conditions for the weak regularity of the semigroup rings are found. A problem is mentioned especially which is much more difficult for weak regularity than for regularity. Only a partial solution of this problem is given for weak regularity.     

Arbitrary vs. regular semigroups

The notion of regularity for semigroups is studied, and it is shown that an unambiguous semigroup (i.e., whose $\text{L}$ and $\text{R}$ orders are respectively unions of disjoint trees) can be embedded in a regular semigroup with the same subgroups and the same ideal structure (except that a zero is added to the regular semigroup).In a previous paper [1] it was shown that any semigroup is the homomorphic image of an unambiguous semigroup with the same groups and a similar ideal structure.Together these two papers thus prove that an arbitrary semigroup divides a regular semigroup with a similar structure.The resulting regular semigroup is finite (resp. torsion, or bounded torsion) if the given semigroup has that property.     

On Topological Semigroups of Matrix Units

On the infinite semigroup of matrix units there exists no semigroup compact [countably compact] topology. Any continuous homomorphism from the infinite topological semigroup of matrix units into a compact topological semigroup is annihilating. The semigroup of matrix units is algebraically h-closed in the class of topological inverse semigroups. Some H-closed minimal semigroup topologies on the infinite semigroup of matrix units are considered.     

On Bruck-Reilly extensions

We present a generalization for the procedure of taking Bruck-Reilly extensions, and we characterize abstractly the regular semigroups which can be obtained in this way. We shall in particular characterize the regular semigroups which can be obtained by considering the usual Bruck-Reilly extensions. Our procedure generalizes Munn’s construction [3] which in its turn combines ideas used by Bruck [1] and Reilly [4].     

On Legal Semigroups

A new class of semigroups with a two variable regularity law is introduced. These semigroups are non-regular semigroups but they are closely related to regular semigroups. The local and global structures of this class of semigroups are investigated.AMS Subject Classification (2000): 20M10Partially supported by a Chinese University of Hong Kong Direct Research grant, Hong Kong (98/99) # 2060152.Partially supported by a grant of the National Science Foundation, China.     

Categorical semigroups

This investigation was stimulated by a question raised by F.R. McMorris and M. Satyanarayana [Proc. Amer. Math. Soc. 33 (1972), 271–277] which asked whether a regular semigroup with a tree of idempotents is categorical. The question is answered in the affirmative. Characterizations of categorical semigroups are found within the following classes of semigroups: regular semigroups, bands, commutative regular semigroups, unions of simple semigroups, semilattices of groups, and commutative semigroups. Some results are related to part of the work of M. Petrich [Trans. Amer. Math. Soc. 170 (1972), 245–268]. For instance, it is shown that the poset of J-classes of any regular categorical semigroup is a tree; however, an example of a regular non-categorical semigroup is given in which the poset of J-classes is a chain. It is also shown that the condition that the subsemigroup of idempotents be categorical is sufficient, but not necessary, for an orthodox semigroup to be categorical.     

保等价部分变换半群的变种半群上的正则元

在现有的保等价部分变换半群的基础上,引入了一个新的运算,得出保等价部分变换半群的变种半群的概念,利用格林关系及幂等元的正则性,讨论了这类半群中元素的正则性,给出了保等价部分变换半群的变种半群中一个元是正则元的充要条件     

Naturally ordered regular semigroups with an inverse monoid transversal

The notion of an inverse transversal of a regular semigroup is well-known. Here we investigate naturally ordered regular semigroups that have an inverse transversal. Such semigroups are necessarily locally inverse and the inverse transversal is a quasi-ideal. After considering various general properties that relate the imposed order to the natural order, we highlight the situation in which the inverse transversal is a monoid. The regularity of Green’s relations is also characterised. Finally, we determine the structure of a naturally ordered regular semigroup with an inverse monoid transversal.     

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     

A Note of Regularity on Completely Distributive Lattices

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Uniform semigroups and fixed point properties

LetS be a uniform semigroup (this includes all topological groups and affine semigroups). We show that a certain space of uniformly continuous functions onS has a left invariant mean iffS has the fixed point property for uniformly continuous affine actions ofS on compact convex sets. This is closely related to but independent of the results of T. Mitchell in [13] and A. Lau in [10]. Interesting examples and consequences are given for the special cases of topological groups and affine convolution semigroups of probability measures on a locally compact semigroup or group. Research Supported by NSERC of Canada Grant No. A8227.     

Countably Compact Paratopological Groups

By means of the theory of bispaces we show that a countably compact T₀ paratopological group (G, τ) is a topological group if and only if (G, τ ∨ τ^-1) is ω-bounded (here τ^-1 is the conjugate topology of τ). Our approach is premised on the fact that every paratopological countably compact paratopological group is a Baire space and on the notion of a 2-pseudocompact space. We also prove that every ω-bounded (respectively, topologically periodic) Baire paratopological group admits a weaker Hausdorff group topology. In particular, ω-bounded (respectively, topologically periodic) 2-pseudocompact (so, also countably compact) paratopological groups enjoy this property. Some topological properties turning countably compact topological semigroups into topological groups are presented and some open questions are posed.     

Uniform distribution and the mean ergodic theorem

Generalizing a result of Veech, [14] Theorem 1.1.5 and answering a question, [14], 1.4 we prove the existence of certain well distributed sequences in topological semigroups having a countable dense subset.     

Orders in rings without identity

G. Lallement [5] proved that every idem potent congruence class of a regular semigroup contains an idem potent. P. Edwards [4] generalized this property of congruences to eventually regular semigroups. Using the natural partial order of the semigroup (see [6]) a weakened version of this result will be proved for the more general class of E-inversive semigroups. But for particular congruences the original result of Lallement still holds for every E-inversive semigroup. Finally, conditions for a congruence on a general semigroup (with E(S) a subsemigroup, resp.) are given, which ensure that Lallement's result holds.     

Topologically complete representations of inverse semigroups

A representation of an inverse semigroup by means of partial open homeomorphisms of a topological T₀-space is called topologically complete if the domains of these partial homeomorphisms form a base of the topology. It is shown how to construct topologically complete representations on the base of a ternary relation satisfying some elementary axioms. This result makes it possible to obtain a pseudo-elementary axiomatization for inverse semigroups that have faithful topologically complete representations in T₁, T₂ and T₃-spaces. A topology is introduced on any antigroup; this topology is a concomitant of the algebraic structure and every topologically complete representation is continuous with respect to this topology.