Semigroups of partial transformations with kernel and image restricted by an equivalence

Semigroup Forum - For an arbitrary set X and an equivalence relation $$mu$$ on X, denote by $$P_mu (X)$$ the semigroup of partial transformations $$alpha$$ on X such that $$xmu subseteq x(ker...     

Semigroups generated by nilpotent transformations

In a seminal paper published in 1966, John Howie characterised the elements of _x, the semigroup (under composition) of all total transformations of a set X into itself, which can be written as a product of idempotents in _x. We now initiate the study of the subsemigroup of _x, the semigroup of all partial transformations of X, which is generated by the nilpotents of _x     

Regularity and Green’s relations on semigroups of transformations with restricted range that preserve an equivalence

Let Y be a subset of X and T(X, Y) the set of all functions from X into Y. Then, under the operation of composition, T(X, Y) is a subsemigroup of the full transformation semigroup T(X). Let E be an equivalence on X. Define a subsemigroup $$T_E(X,Y)$$ of T(X, Y) by $$\begin{aligned} T_E(X,Y)=\{\alpha \in T(X,Y):\forall (x,y)\in E, (x\alpha ,y\alpha )\in E\}. \end{aligned}$$Then $$T_E(X,Y)$$ is the semigroup of all continuous self-maps of the topological space X for which all E-classes form a basis carrying X into a subspace Y. In this paper, we give a necessary and sufficient condition for $$T_E(X,Y)$$ to be regular and characterize Green’s relations on $$T_E(X,Y)$$. Our work extends previous results found in the literature.     

Orbit equivalence of p-adic transformations and their iterates

Orbit equivalence is a weaker notion of equivalence than isomorphism for measurable systems. We examine \(p\) -adic transformations in various orbit equivalence classes, including an example that preserves an infinite measure. Although \(p\) -adic transformations have been studied with respect to Haar measure, we use other i.i.d. product measures to see examples of different orbit equivalence classes. Since translation by an integer is an iterate of translation by 1, we gain an understanding of the possible behaviors of orbit equivalence classes under iteration.     

Higher-order Darboux transformations with foreign auxiliary equations and equivalence with generalized Darboux transformations

We show that a recently developed modified Darboux transformation that uses foreign auxiliary equations, can be unified with the Darboux transformation for generalized Schrödinger equations. As a consequence of this unification, we obtain explicit Darboux transformations with foreign auxiliary equations of arbitrary order.     

一类微分-差分方程组的内禀对称和等价群变换

研究一类微分-差分方程组的对称和等价群变换.采取内禀的无穷小算子方法,给出了方程组的内禀对称和等价群变换.为结合抽象Lie代数结构,给方程完全分类提供了理论基础.     

关于反射等价关系的变换半群的注记

设TX是非空集合X上全变换半群,E是X上非平凡的等价关系,则T?(X)是TX的子半群.在赋予半群T?(X)自然偏序关系的条件下,本文刻画了它的相容元.     

Semigroups generated by a group and an idempotent

It is well known that the semigroup of all transformations on a finite set X of order n is generated by its group of units, the symmetric group, and any idempotent of rank n ? 1. Similarly, the symmetric inverse semigroup on X is generated by its group of units and any idempotent of rank n ? 1 while the analogous result is true for the semigroup of all n × n matrices over a field.

In this paper we begin a systematic study of the structure of a semigroup S generated by its group G of units and an idempotent ? . The first section consists of preliminaries while the second contains some general results which provide the setting for those which follow.

In the third section we shall investigate the situation where G is a permutation group on a set X of order n and ? is an idempotent of rank n ? 1. In particular, we shall show that any such semigroup S is regular. Furthermore we shall determine when S is an inverse or orthodox semigroup or completely regular semigroup.

The fourth section deals with a special case, that in which G is cyclic. The fifth, and last, deals with the situation where G is dihedral. In both cases, the resulting semigroup has a particularly delicate structure which is of interest in its own right. Both situations are replete with interesting combinatorial gems.

The author was led to the results of this paper by considering the output of a computer program he was writing for generating and analyzing semigroups.     

Admissible equivalence transformations for linearization of nonlinear wave type equations

In the present paper, we consider a general family of two‐dimensional wave equations, which represents a great variety of linear and nonlinear equations within the framework of the transformations of equivalence groups. We have investigated the existence problem of point transformations that lead mappings between linear and nonlinear members of particular families and determined the structure of the nonlinear terms of linearizable equations. We have also given examples about some equivalence transformations between linear and nonlinear equations and obtained exact solutions of nonlinear equations via the linear ones.     

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.     

On the equivalence of the BLUEs under a general linear model and its restricted and stochastically restricted models

Necessary and sufficient conditions are derived for the equalities of the best linear unbiased estimators (BLUEs) of parametric functions under a general linear model and its restricted and stochastically restricted models to hold.     

The regular part of a semigroup of transformations with restricted range

Let T(X) be the full transformation semigroup on the set X and let T(X,Y) be the semigroup consisting of all total transformations from X into a fixed subset Y of X. It is known that $$F(X, Y)=\{\alpha\in T(X, Y): X\alpha\subseteq Y\alpha\},$$ is the largest regular subsemigroup of T(X,Y) and determines Green??s relations on T(X,Y). In this paper, we show that F(X,Y)?T(Z) if and only if X=Y and |Y|=|Z|; or |Y|=1=|Z|, and prove that every regular semigroup S can be embedded in F(S ¹,S). Then we describe Green??s relations and ideals of F(X,Y) and apply these results to get all of its maximal regular subsemigroups when Y is a nonempty finite subset of X.     

A note on abundance of certain semigroups of transformations with restricted range

Given a set X and a nonempty Y?X, we denote by T(X,Y) the subsemigroup of the full transformation semigroup on X consisting of all transformations whose range is contained in Y. We show that the semigroup T(X,Y) is right abundant but not left abundant whenever Y is a proper non-singleton subset of X.     

Unary Semigroups with an Associate Subgroup

A subgroup H of a regular semigroup S is said to be an associate subgroup of S if for every s ∈ S, there is a unique associate of s in H. An idempotent z of S is said to be medial if czc = c, for every c product of idempotents of S. Blyth and Martins established a structure theorem for semigroups with an associate subgroup whose identity is a medial idempotent, in terms of an idempotent generated semigroup, a group and a single homomorphism. Here, we construct a system of axioms which characterize these semigroups in terms of a unary operation satisfying those axioms. As a generalization of this class of semigroups, we characterize regular semigroups S having a subgroup which is a transversal of a congruence on S.     

Automorphisms that admit an approximation by periodic transformations

Summary We are concerned with a class of automorphisms which can be approximated by periodic transformations. We show that such an automorphism T is ergodic and that a relationship exists between the spectrum of T, the measure of the sets which are invariant under T, and the ergodicity of the powers of T. Finally, we exhibit conditions under which T is weakly mixing and not strongly mixing.This research was supported by NSF Grant No. GP-7490.