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.     

Semitransitive subsemigroups of the symmetric inverse semigroups

We initiate the study of semitransitive transformation semigroups. In the paper we describe the structure of semitransitive subsemigroups of the finite symmetric inverse semigroup of the minimal cardinality modulo the classification of transitive subgroups of the minimal cardinality of finite symmetric groups, and state the results on minimal transitive subsemigroups. The authors were supported in part by Ukrainian-Slovenian bilateral research grants from the Ministry of Education and Science, Ukraine, and the Research Agency of the Republic of Slovenia.     

Maximal properties of some subsemigroups in finite order-preserving transformation semigroups

Let [n] = {1,2,…,n} be a finite set, ordered in the usual way. The order-preserving transformation semigroup O_n is the set of all order-preserving transformations of [n] (excluding the identity mapping) under composition. In this paper we first describe maximal idempotent-generated subsemigroups of O _n, and show that O_n has 2n - 2 such subsemi-groups. Secondly, we investigate maximal regular subsemigroups of O_n , and obtain the number of such subsemigroups as 2n - 3. Thirdly, we describe maximal idempotent-generated regular subsemigroups of O_n , and also obtain their classification and number.     

真理想为Archimedean子半群的偏序半群

根据真理想情况给出了偏序半群的一种分类，研究了真理想为Archimedean子半群的偏序半群的特征．     

Frucht theorem for inverse semigroups

In the paper, the problem of representing a finite inverse semigroup by partial transformations of a graph is treated. The notions of weighted graph and its weighted partial isomorphisms are introduced. The main result is that any finite inverse semigroup is isomorphic to the semigroup of weighted partial isomorphisms of a weighted graph. This assertion is a natural generalization of the Frucht theorem for groups. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 246–251, February, 1997. This research was partially supported by the International Science Foundation under grant No. GSU 041049. Translated by A. I. Shtern     

Cayley color graphs of inverse semigroups and groupoids

The notion of Cayley color graphs of groups is generalized to inverse semigroups and groupoids. The set of partial automorphisms of the Cayley color graph of an inverse semigroup or a groupoid is isomorphic to the original inverse semigroup or groupoid. The groupoid of color permuting partial automorphisms of the Cayley color graph of a transitive groupoid is isomorphic to the original groupoid.     

A relation for general and inverse semigroups

The relation in the title is S defined by

A classification of maximal subsemigroups of finite order-preserving transformation semigroups

We describe the maximal subsemigroups of the semigroup of all order-preserving transformations of a finite chain and completely obtain their classification. We also count the number of its maximal subsemigroups.     

Inverse subsemigroups and classes of finite aperiodic semigroups

We prove a number of results related to finite semigroups and their inverse subsemigroups, including the following. (1) A finite semigroup is aperiodic if and only if it is a homomorphic image of a finite semigroup whose inverse subsemigroups are semilattices. (2) A finite inverse semigroup can be represented by order-preserving mappings on a chain if and only if it is a semilattice. Finally, we introduce the concept of pseudo-small quasivariety of finite semigroups, generalizing the concept of small variety.     

0-范畴E-酉逆半群的结构

本证明了每一个0-范畴E-酉逆半群能被嵌入到半格与本原逆半群的λ一半直积中。     

Dense inverse subsemigroups of a topological inverse semigroup

In this paper we study dense inverse subsemigroups of topological inverse semigroups. We construct a topological inverse semigroup from a semilattice. Finally, we give two examples of the closure of B _{( −∞, ∞ )}¹, a topological inverse semigroup obtained by starting with the real numbers as a semilattice with the operation a ∨ b=sup{a,b}. The author would like to thank to the referee for useful suggestions.     

Locally maximal idempotent-generated subsemigroups of singular orientation-preserving transformation semigroups

In this paper we study locally maximal idempotent-generated subsemigroups of finite singular orientation-preserving transformation semigroups. This work is supported by N.S.F. and E.S.F. of Guizhou.     

A construction of weakly inverse semigroups

Let S° be an inverse semigroup with semilattice biordered set E° of idempotents and E a weakly inverse biordered set with a subsemilattice Ep = { e ∈ E ｜ arbieary f ∈ E, S（f , e） loheain in w（e）} isomorphic to E° by θ：Ep→E°. In this paper, it is proved that if arbieary f, g ∈E, f ←→ g→→ f°θD^s° g°θand there exists a mapping φ from Ep into the symmetric weakly inverse semigroup P J（E∪ S°） satisfying six appropriate conditions, then a weakly inverse semigroup ∑ can be constructed in P J（S°）, called the weakly inverse hull of a weakly inverse system （S°, E, θ, φ） with I（∑） ≌ S°, E（∑） ∽- E. Conversely, every weakly inverse semigroup can be constructed in this way. Furthermore, a sufficient and necessary condition for two weakly inverse hulls to be isomorphic is also given.     

Semigroup closures of finite rank symmetric inverse semigroups

We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse semigroup of finite transformations I _λ ⁿ of the rank ≤ n is algebraically closed in the class of (semi)topological inverse semigroups with continuous inversion. We also derive related results about the nonexistence of (partial) compactifications of classes of semigroups that we consider.     

A classification of maximal idempotent-generated subsemigroups of singular orientation-preserving transformation semigroups

We describe maximal idempotent-generated subsemigroups of finite singular orientation-preserving transformation semigroups and completely obtain their classification. This work is supported by N.S.F. and E.S.F. of Guizhou.     

Ring semigroups whose subsemigroups form a chain

A multiplicative semigroup S is called a ring semigroup if an addition may be defined on S so that (S,+,⋅) is a ring. Such semigroups have been well-studied in the literature (see Bell in Words, Languages and Combinatorics, pp. 24–31, World Scientific, Singapore, 1994; Jones in Semigroup Forum 47(1):1–6, 1993; Jones and Ligh in Semigroup Forum 17(2):163–173, 1979). In this note, we use Mihăilescu’s Theorem (formerly Catalan’s Conjecture) to characterize the ring semigroups whose subsemigroups containing 0 form a chain with respect to set inclusion.     

Cyclic subsemigroups of symmetric inverse semigroups

A generalization of the cycle notation for permutations is introduced for partial one-one transformations (charts). Notational representation theorems for charts that generalize those of permutations are given. Notational multiplication of charts is developed and then applied to yield a transparent proof of Frobenius' result which bounds the idempotent in the cyclic subsemigroup. Lastly, the well known result that the structure of the cyclic subgroups of the finite symmetric groups is determined from combinations of disjoint cycles is generalized to the cyclic subsemigroups of the finite symmetric inverse semigroups.     

Inverse semigroups determined by their lattices of convex inverse subsemigroups I

Every inverse semigroup possesses a natural partial order and therefore convexity with respect to this order is of interest. We study the extent to which an inverse semigroup is determined by its lattice of convex inverse subsemigroups; that is, if the lattices of two inverse semigroups are isomorphic, how are the semigroups related? We solve this problem completely for semilattices and for inverse semigroups in general reduce it to the case where the lattice isomorphism induces an isomorphism between the semilattices of idempotents of the semigroups. For many inverse semigroups, such as the monogenic ones, this case is the only one that can occur. In Part II, a study of the reduced case enables us to prove that many inverse semigroups, such as the free ones, are strictly determined by their lattices of convex inverse subsemigroups, and to show that the answer obtained here for semilattices can be extended to a broad class of inverse semigroups, including all finite, aperiodic ones. Received September 24, 2002; accepted in final form December 15, 2002.     

Restricted algebras on inverse semigroups I,representation theory

The relation between representations and positive definite functions is a key concept in harmonic analysis on topological groups. Recently this relation has been studied on topological groupoids. This is the first in a series of papers in which we have investigated a similar relation on inverse semigroups. We use a new concept of “restricted” representations and study the restricted semigroup algebras and corresponding C *‐algebras. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Soft ordered semigroups

Molodtsov introduced 1999 the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. In this paper we apply the notion of soft sets by Molodtsov to ordered semigroups. The notions of (trivial, whole) soft ordered semigroup, soft ordered subsemigroup, soft left (right) ideal, and left (right) idealistic soft ordered semigroup are introduced, and various related properties are investigated (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)