The minimal number of generators of a finite semigroup

The rank of a finite semigroup is the smallest number of elements required to generate the semigroup. A formula is given for the rank of an arbitrary (not necessarily regular) Rees matrix semigroup over a group. The formula is expressed in terms of the dimensions of the structure matrix, and the relative rank of a certain subset of the structure group obtained from subgroups generated by entries in the structure matrix, which is assumed to be in Graham normal form. This formula is then applied to answer questions about minimal generating sets of certain natural families of transformation semigroups. In particular, the problem of determining the maximum rank of a subsemigroup of the full transformation monoid (and of the symmetric inverse semigroup) is considered.     

Matrix rank 1 semigroup identities

A finite basis of identities is constructed for the semigroup of all rank 1 n × n matri­ces over the field. It is worthy to notice that every semigroup of all rank r, r > l,n×n matrices over a finite field has no finite basis of identities. Let G be an arbitrary vari­ety of groups with a finite basis of identities. A finite basis of identities is constructed for the variety generated by all completely 0-simple semigroups over G-groups.     

Structure of a permutable Munn semigroup of finite rank

A semigroup any two congruences of which commute as binary relations is called a permutable semigroup. We describe the structure of a permutable Munn semigroup of finite rank. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 742–746, June, 2006.     

The semigroup generated by a unitary orbit of a singular matrix

We investigate the structure of the multiplicative semigroup generated by the set of matrices that are unitarily equivalent to a given singular matrix A. In particular, we give necessary and sufficient conditions, in terms of the singular values of A, for such a semigroup to consist of all matrices of rank not exceeding the rank of A.     

The semigroup generated by a unitary orbit of a singular matrix

We investigate the structure of the multiplicative semigroup generated by the set of matrices that are unitarily equivalent to a given singular matrix A. In particular, we give necessary and sufficient conditions, in terms of the singular values of A, for such a semigroup to consist of all matrices of rank not exceeding the rank of A.     

P-完全正则半群

求证具有强半格结构的完全正则半群成为P-完全正则半群的充分条件.利用半群的强半格结构以及同余的性质.完全单半群的强半格-正规群带是P-完全正则半群.矩形群的强半格正规纯正群类ONBG,左群的强半格左正规纯整群类LONBG,群的强半格Clifford半群类,矩形带的强半格正规带类NB,都具有性质P.     

The rank of the inverse semigroup of partial automorphisms on a finite fence

Semigroup Forum - A fence is a particular partial order on a (finite) set, close to the linear order. In this paper, we calculate the rank of the semigroup $$\mathscr {FI}_{n}$$ of all...     

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.     

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.     

Clifford半群的E-扩张

半群扩张是以已知的或者较简单的半群类为基础,借助某种方法扩张成一类半群.本文讨论Cliffbrd半群借助Mum半群TE的扩张.引入了这种扩张的概念并给出一个具体的例子.还给出Clifford半群有E扩张的必要条件.证明了满逆半群的Green关系等价类在Munn表示下的同态象仍是同样的等价类.     

I_(E*)(X)中E类方向保序变换半群的秩

设X为有限集合,E为X上的等价关系.令OPPI_E*(X)为所有I_R*(X)中E类方向保序变换构成的半群.在一定的条件下讨论了OPPI_E*(X)的Green关系与秩.     

Every semigroup is isomorphic to a transitive semigroup of binary relations

Every (finite) semigroup is isomorphic to a transitive semigroup of binary relations (on a finite set).

     

Free local semigroup constructions

We introduce the notions of causal paths and causal homotopies, modifications of the traditional notions of paths and homotopies, as more suitable for certain basic constructions in (Lie) semigroup theory. The major result is the construction in this causal context of an analogue of the universal covering semigroup and the demonstration that local homomorphisms on the given semigroup extend to global homomorphisms on it. In certain important cases, it is shown that this semigroup actually agrees with the universal covering semigroup.The author gratefully acknowledges the partial support of theNational Science Foundation. DMS 910-4582.     

半群的i-v Fuzzy理想

在半群上引入i-v Fuzzy理想的概念.研究了半群的i-v Fuzzy理想的若干性质,特别是给出了半群的i-v Fuzzy子集成为理想的若干特征性质.     

Coefficient rings of numerical semigroup algebras

Numerical semigroup rings are investigated from the relative viewpoint. It is known that algebraic properties such as singularities of a numerical semigroup ring are properties of a flat numerical semigroup algebra. In this paper, we show that arithmetic and set-theoretic properties of a numerical semigroup ring are properties of an equi-gcd numerical semigroup algebra.

     

The large rank of a finite semigroup using prime subsets

The large rank of a finite semigroup \(\Gamma \) , denoted by \(r_5(\Gamma )\) , is the least number \(n\) such that every subset of \(\Gamma \) with \(n\) elements generates \(\Gamma \) . Howie and Ribeiro showed that \(r_5(\Gamma ) = |V| + 1\) , where \(V\) is a largest proper subsemigroup of \(\Gamma \) . This work considers the complementary concept of subsemigroups, called prime subsets, and gives an alternative approach to find the large rank of a finite semigroup. In this connection, the paper provides a shorter proof of Howie and Ribeiro’s result about the large rank of Brandt semigroups. Further, this work obtains the large rank of the semigroup of order-preserving singular selfmaps.     

Structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental

We describe the structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 52–60, January, 2009.     

Identities of semigroup algebras of completely O-simple semigroups

Let H = M⁰(G; I, ; P) be a Rees semigroup of matrix type with sandwich matrix P over a group H⁰ with zero. If F is a subgroup of G of finite index and X is a system of representatives of the left cosets of F in G, then with the matrix P there is associated in a natural way a matrix P(F, X) over the group F⁰ with zero. Our main result: the semigroup algebra K[H] of H over a field K of characteristic 0 satisfies an identity if and only if G has an Abelian subgroup F of finite index and, for any X, the matrix P(F, X) has finite determinant rank.Translated from Matematicheskie Zametki, Vol. 18, No. 2, pp. 203–212, August, 1975.     

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     

Some examples of semigroup algebras of finite representation type

The semigroup algebras over a field K of the semigroups T_n of all permutations of a set of n elements are considered. It is proved: if n≤3 and (n!)^-1∈ K then the algebra KT_n has a finite representation type. Also the finiteness of the representation type of the semigroup algebra KS is established, where S is the sub-semigroup of T_n (n is arbitrary) such that S=J_n∪G where J_n={x∈T_n|rank x=1}, while G is a doubly transitive subgroup of the symmetric group S_n, the order of G being invertible in K. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 229–238, 1987.