Maximal Subsemigroups of Finite Transformation Semigroups K（n, r）

Let Tn be the full transformation semigroup on the n-element set Xn. For an arbitrary integer r such that 2 ≤ r ≤ n-1, we completely describe the maximal subsemigroups of the semigroup K(n, r) = {α∈Tn : |im α| ≤ r}. We also formulate the cardinal number of such subsemigroups which is an answer to Problem 46 of Tetrad in 1969, concerning the number of subsemigroups of Tn.     

Rank Properties of the Semigroup of Singular Transformations on a Finite Set

It is known that the semigroup Sing _n of all singular self-maps of X _n  = {1,2,…, n} has rank n(n ? 1)/2. The idempotent rank, defined as the smallest number of idempotents generating Sing _n , has the same value as the rank. (See Gomes and Howie, 1987 Gomes , G. M. S. , Howie , J. M. ( 1987 ). On the rank of certain finite semigroups of transformations . Math. Proc. Cambridge Phil. Soc. 101 : 395 – 303 .[Crossref], [Web of Science ®] , [Google Scholar].) Idempotents generating Sing _n can be seen as special cases (with m = r = 2) of (m, r)-path-cycles, as defined in Ay\i k et al. (2005 Ay?k , G. , Ay?k , H. , Howie , J. M. ( 2005 ). On factorisations and generators in transformation semigroups . Semigroup Forum 70 : 225 – 237 .[Crossref], [Web of Science ®] , [Google Scholar]). The object of this article is to show that, for fixed m and r, the (m, r)-rank of Sing _n , defined as the smallest number of (m, r)-path-cycles generating Sing _n , is once again n(n ? 1)/2.     

The Monoid of All Injective Order Preserving Partial Transformations on a Finite Chain

In this paper we study several structural properties of the monoids \poi _n of all injective order preserving partial transformations on a chain with n elements. Our main aim is to give a presentation for these monoids. January 27, 1999     

Quasi-duality for the rings of generalized power series *

Let A, B be associative rings with identity, and (S, ≤) a strictly totally ordered commutative monoid which is also artinian. For any bimodule _AM_B , we construct a bimodule _A[[S]]M[S]_B[[S]] and prove that _AM_B defines a quasi-duality if and only if the bimodule _A[[S]]M[S]_B[[S]] defines a quasi-duality. As a corollary, it is shown that if a ring A has a quasi-duality then the ring A[[S]] of generalized power series over A has a quasi-duality.     

Maximal Inverse Subsemigroups of the Symmetric Inverse Semigroup on a Finite-Dimensional Vector Space

The notion of “near isomorphism” for torsion-free Abelian groups of finite rank is well known. In particular, this concept turned out to be of importance for classifying almost completely decomposable groups. We extend near isomorphism to classes of torsion-free Abelian groups of infinite rank which are unions of bcd–groups, this is to say unions of groups which are bounded essential extensions of completely decomposable groups. Moreover, we show that nearly isomorphic groups of this class also have nearly isomorphic endomorphism rings considered as Abelian groups.     

The Ideal Structure of Semigroups of Linear Transformations with Upper Bounds on Their Nullity or Defect

Suppose V is a vector space with dim V = p ≥ q ≥ ?₀, and let T(V) denote the semigroup (under composition) of all linear transformations of V. For α ∈ T (V), let ker α and ran α denote the “kernel” and the “range” of α, and write n(α) = dim ker α and d(α) = codim ran α. In this article, we study the semigroups AM(p, q) = {α ∈ T(V):n(α) < q} and AE(p, q) = {α ∈ T(V):d(α) < q}. First, we determine whether they belong to the class of all semigroups whose sets of bi-ideals and quasi-ideals coincide. Then, for each semigroup, we describe its maximal regular subsemigroup, and we characterise its Green's relations and (two-sided) ideals. As a precursor to further work in this area,, we also determine all the maximal right simple subsemigroups of AM(p, q).     

Asymptotic Results for Semigroups of Order-Preserving Partial Transformations

Let 𝒫  _n be the semigroup of all decreasing and order-preserving partial transformations of an n-element chain, and let E(𝒫  _n ) be its set of idempotents. Among other results, asymptotic formulae for |𝒫  _n | and |E(𝒫  _n )|/|𝒫  _n | are obtained. Similar results for 𝒫  _n the (larger) semigroup of all order-preserving partial transformations of an n-element chain are also obtained.     

Idempotent Generation in the Endomorphism Monoid of a Uniform Partition

Denote by 𝒯_n and 𝒮_n the full transformation semigroup and the symmetric group on the set {1,…, n}, and ?_n = {1} ∪ (𝒯_n?𝒮_n). Let 𝒯(X, 𝒫) denote the monoid of all transformations of the finite set X preserving a uniform partition 𝒫 of X into m subsets of size n, where m, n ≥ 2. We enumerate the idempotents of 𝒯(X, 𝒫), and describe the submonoid S = ? E ? generated by the idempotents E = E(𝒯(X, 𝒫)). We show that S = S₁ ∪ S₂, where S₁ is a direct product of m copies of ?_n, and S₂ is a wreath product of 𝒯_n with 𝒯_m?𝒮_m. We calculate the rank and idempotent rank of S, showing that these are equal, and we also classify and enumerate all the idempotent generating sets of minimal size. In doing so, we also obtain new results about arbitrary idempotent generating sets of ?_n.     

OI_n的理想K(n,r)的极大逆子半群

Xn为n元有限集,OIn为Xn上的一切保序严格部分一一变换半群.记K(n,r)={α∈OIn∶|Tmα|≤r}(0≤r≤n-1)则K(n,r)(0≤r≤n-1)是OIn的理想.我们刻划了K(n,r)(1≤r≤n-1)的极大逆子半群.     

Abundance of Semigroups of Transformations Restricted by an Equivalence

Let σ be an equivalence on X, and let E(X, σ) denote the semigroup (under composition) of all f: X → X such that σ ? ker(f). In this article, we show that the semigroup E(X, σ) is right abundant but not left abundant whenever |X| ≥3 and σ is non-trivial.     

Factorisable Semigroups of Generalized Transformations

ABSTRACT

If X and Y are sets, we let P(X, Y ) denote the set of all partial transformations from X into Y (that is, all mappings whose domain and range are subsets of X and Y, respectively). We define an operation * on P(X, Y ) by choosing θ ∈ P(Y, X) and writing: α*β = α °θ°β, for each α, β ∈ P(X, Y ). Then (P(X, Y ), *) is a semigroup, and some authors have determined when this is regular (Magill and Subbiah, 1975 Magill , K. D. , Jr. Subbiah , S. ( 1975 ). Green's relations for regular elements of sandwich semigroups. I. General results . Proc. London Math. Soc. 31 : 194 – 210 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), when it contains a “proper dense subsemigroup” (Wasanawichit and Kemprasit, 2002 Wasanawichit , A. , Kemprasit , Y. ( 2002 ). Dense subsemigroups of generalized transformation semigroups . J. Austral. Math. Soc. 73 ( 3 ): 433 – 445 . [CSA] [Crossref] , [Google Scholar]) and when it is factorisable (Saengsura, 2001 Saengsura , K. ( 2001 ). Factorizable on (P(X, Y ), θ) , MSc thesis, 23 pp (in Thai, with English summary), Department of Mathematics, Khon Kaen University, Khon Kaen, Thailand, 2001.  [Google Scholar]). In this paper, we extend the latter work to certain subsemigroups of (P(X, Y ), *). We also consider the corresponding idea for partial linear transformations from one vector space into another. In this way, we generalise known results for total transformations and for injective partial transformations between sets, and we establish new results for linear transformations between vector spaces.     

Second Centralizers of Partial Transformations

Second centralizers of partial transformations on a finite set are determined. In particular, it is shown that the second centralizer of any partial transformation consists of partial transformations that are locally powers of .     

可拓积分的极限定理

在可拓集合论、可拓测度论的基础上给出了可拓积分的概念，讨论了它的三个性质，并研究了可测函数的可拓积分的极限定理．     

Generating Sets of Finite Transformation Semigroups PK(n,r) and K (n,r)

Let P _n and T _n be the partial transformation and the full transformation semigroups on the set {1,…, n}, respectively. In this paper we find necessary and sufficient conditions for any set of partial transformations of height r in the subsemigroup PK(n, r) = {α ∈P _n : |im (α)| ≤r} of P _n to be a (minimal) generating set of PK(n, r); and similarly, for any set of full transformations of height r in the subsemigroup K(n, r) = {α ∈T _n : |im (α)| ≤r} of T _n to be a (minimal) generating set of K(n, r) for 2 ≤ r ≤ n ? 1.     

Presentations for Two Extensions of the Monoid of Order-Preserving Mappings on a Finite Chain

The purpose of this paper is to give presentations for the monoids of orientation-preserving mappings on a finite chain of order n, and orientation-preserving or reversing mappings on such a chain. Both these monoids are natural extensions of the monoid of order-preserving mappings. The obtained presentations are on two and three generators, respectively, and have n + 2 and n + 6 relations, respectively.AMS Subject Classification (1991): 20M05 20M20 06A05     

On the Number of Nilpotents in the Partial Symmetric Semigroup

Abstract

In this note, we obtain and discuss formulae for the total number of nilpotent partial and nilpotent partial one–one transformations of a finite set.     

保持两个等价关系的变换半群的Green关系

Let Tx be the full transformation semigroup on a set X. For a non-trivial equivalence F on X, let
TF（X） = {f ∈ Tx ： arbieary （x, y） ∈ F, （f（x）,f（y）） ∈ F}.
Then TF（X） is a subsemigroup of Tx. Let E be another equivalence on X and TFE（X） = TF（X） ∩ TE（X）. In this paper, under the assumption that the two equivalences F and E are comparable and E lohtain in F, we describe the regular elements and characterize Green＇s relations for the semigroup TFE（X）.     

奇异变换半群Sing_n的非群平方幂等元秩

设Sing_n是[n]上的奇异变换半群.证明了半群Sing_n是由秩为n-1的非群平方幂等元生成的,且它的非群平方幂等元秩为(n(n-1))/2.     

On the Ranks of Certain Semigroups of Orientation Preserving Transformations

We classify the unmixed squarefree lexsegment ideals and determine those which are Cohen–Macaulay.