On Finite Semigroups Embeddable in Inverse Semigroups

No Abstract. October 29, 1998     

Generalized Bicyclic Semigroups and Jones Semigroups

In this paper, we consider the generalized bicyclic semigroups B_n = a, b | aⁿb = 1 and the Jones semigroups A_n = a, b | aⁿ⁺¹b = a. They are the generalizations of the bicyclic semigroup B = a, b | ab = 1 and its analogous semigroup A = a, b | a²b = a discovered by P.R., Jones in 1987. The word problem for these kinds of semigroups is solved. It is proved that, for n 2, B_n are bisimple right inverse but not inverse semigroups and that the semigroup C = a, b | a²b = a, ab² = b is the smallest idempotent-free homomorphic image of A_n. Moreover, we also prove that A_n and A_m are mutually embeddable but not isomorphic with each other if n m. As a consequence, different kind of -nontrivial [0-]simple semigroups without idempotents are discussed.AMS 1991 Subject Classification: primary 20M10 secondary 20M05.Supported by NNSF of China (19671063) and KSRF of Sichuan Education Committee ([1999]127).     

广义双循环半群和Jones半群

本文刻画了广义双循环半群Ｂｎ＝〈ａ,ｂ｜ａ＾ｎｂ＝１〉和Ｊｏｎｅｓ半群Ａｎ＝〈ａ,ｂ｜ａ＾ｎ＋１ｂ＝ａ〉（ｎ≥１）的结构;证明了每个Ａｎ都具有Ｐ．Ｒ．Ｊｏｎｅｓ所发现的半群Ａ＝〈ａ,ｂ｜ａ＾２ｂ＝ａ〉的所有重要性质,特别地,证明了Ａｎ,Ａｍ可互相嵌入,从而得到：第三个Ｄ－非平凡的无幂等元「０－」单半群若不含Ｃ＝〈ａ,ｂ｜ａ＾２ｂ＝ａ,ａｂＴ＾２＝ｂ〉,则必含每个Ａｎ或它们的对偶,作为推论,每人广义     

Cleavability in Semigroups

The notion of cleavability (splittability) is observed to apply not only to topological spaces, where it was first developed, but to semigroups also. Several results of the form 'if a semigroup D is cleavable over a class of semigroups each of which has property P, then D also has property P' are derived, and some suggestions for further investigations are put forward.     

Pseudo-similarity in Semigroups

In this paper, we get some new characterizations of pseudo-similarity for semigroups. As an application, we deduce that if R is a QB-ring then $a \barsim b$ in R if and only if for all $n \memb N$, there exists $u \memb$ R^-1 _q such that aⁿ = ubⁿu^-1 _q and bⁿ = u^-1 _q aⁿu.     

Semigroups with I-Quasi Length and Syntactic Semigroups

A module M is said to be extending (𝒢-extending) if for each submodule X of M there exists a direct summand D of M such that X is essential in D (X ∩ D is essential in both X and D). It is known that for a nonsingular module the concepts of 𝒢-extending and extending coincide. However, in the not nonsingular case, they are distinct. In this article, we obtain a characterization of the right 𝒢-extending generalized triangular matrix rings. This result and its corollaries improve and generalize the existing results on right extending generalized triangular matrix rings. It is well known that the ring of n-by-n triangular matrices over a right selfinjective ring is not, in general, right extending. One application of our characterization shows that such rings are right 𝒢-extending. Connections to Operator Theory and a characterization of the class of right extending right SI-rings are also obtained. Examples are given to illustrate and delimit the theory.     

完全单半群及完全正则半群的逆断面

指出完全单半群Ｓ的任何一个Ｆ－类是逆断面,且为Ｑ－逆断面,而Ｓ的任何一个逆断面必是一个Ｆ－类,因而所有逆断面同构。并且给出完全正则半群的逆断面存在的充要条件。     

Two-Parameter Semigroups

In this paper we study compact semigroups that can be written as a product of two one-parameter semigroups. We show that such semigroups admit locally a unique factorization and can locally be embedded into a two-dimensional Lie group. The latter fact allows us to determine their local structure. We show how these results provide generalizations of a number of earlier results of this type.     

Annihilators in Nonrepetitive Semigroups

It is well known that one can construct an infinite sequence on three or more symbols such that no two consecutive blocks are identical. In this paper we give a combinatorial characterization for those finite sequences which are maximal in the sense that appending to them even one more symbol yields two consecutive identical blocks.     

Power Semigroups of a Class of Cliford Semigroups

Let S =∪(Gα : α∈ E) be a semilattice of groups(i.e., a Cliford semigroup) and n a natural number. E is called an n-element chain of groups if it is an n-element chain. Denote by Cn the set of all n-element chains of groups. In this paper we shall show that for any natural number n, the class of semigroups Cn satisfies the strong isomorphism property.     

Embedding of Countable Topological Semigroups in Simple Countable Connected Topological Semigroups

We prove that any countable Hausdorff topological (inverse) semigroup is topologically isomorphically embedded into a simple countable connected Hausdorff topological (inverse) semigroup with identity.     

Permanent Semigroups

A permanent semigroup is a semigroup of n × n matrices on which the permanent function is multiplicative. If the underlying ring is an infinite integral domain with characteristic p > n or characteristic 0 we prove that any permanent semigroup consists of matrices with at most one nonzero diagonal. The same result holds if the ring is a finite field with characteristic p > n and at least n²+n elements. We also consider the Kronecker product of permanent semigroups and show that the Kronecker product of permanent semigroups is a permanent semigroup if and only if the pennanental analogue of the formula for the determinant of a Kronecker product of two matrices holds. This latter result holds even when the matrix entries are from a commutative ring with unity.     

Idempotents in n-ary Semigroups

We select a new type of elements of n-ary (n 3) semigroups with an idempotent and investigate properties of ideals connected with these elements.AMS Subject Classification (2000) 20N15     

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.     

Unary Iterative Hyperidentities for Semigroups and Inverse Semigroups

Research supported by NSERC.     

Metrized Semigroups

The notion of metrized order (antimetric) on a topological group is characterized by three equivalent systems of axioms and connected with pointed locally generated semigroups. In the present paper, these notions are discussed and new results are announced. The main result is an analog of the following fact in metric geometry: every left-invariant inner metric on a Lie group is Finsler (maybe, nonholonomic). In the situation considered, a norm is replaced by an antinorm, and a metric by an antimetric. Examples are given, showing the complexity of these structures and their prevalence. Among them are: a nonholonomic antimetric on Heisenberg group, an antimetric on a nonnilpotent group admitting dilatations, a pointed locally generated semigroup in the Hilbert space with trivial tangent cone, antinorms connected with the Brunn–Minkowski inequality and Shannon entropy, a discontinuous antinorm on a Lie algebra determining a continuous antimetric on the Lie group, and an example of the converse situation. Problems are formulated. Bibliography: 47 titles.     

半群中的粗模糊理想

将粗糙集的理论方法用于模糊半群的研究。首先,给出上(下)粗模糊子半群及上(下)粗模糊理想等概念,并研究了它们的有关性质。随后,证明了上粗模糊子半群与上粗模糊理想是通常的模糊子半群与模糊理想概念的扩张。特别地,运用粗糙集的思想方法研究半群上的模糊同余及其截关系,给出它们的一些基本性质。     

Arithmetic Processes in Semigroups

We consider the asymptotic behavior of the distributions of stochastic processes defined by multiplicative functions in an arithmetic semigroup.     

Constructing Almost Symmetric Numerical Semigroups from Irreducible Numerical Semigroups

Every almost symmetric numerical semigroup can be constructed by removing some minimal generators from an irreducible numerical semigroup with its same Frobenius number.