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Semigroup Forum

, Volume 71, Issue 2, pp 241–251 | Cite as

Green's Equivalences on Semigroups of Transformations Preserving Order and an Equivalence Relation

  • Pei HuishengEmail author
  • Zou Dingyu
Article

Abstract

Let ${\cal T}_X$ be the full transformation semigroup on the set $X$, \[ T_{E}(X)=\{f\in {\cal T}_X\colon \ \forall(a,b)\in E,(f(a),f(b))\in E\} \] be the subsemigroup of ${\cal T}_X$ determined by an equivalence $E$ on $X$. In this paper the set $X$ under consideration is a totally ordered set with $mn$ points where $m\geq 2$ and $n\geq 3$. The equivalence $E$ has $m$ classes each of which contains $n$ consecutive points. The set of all order preserving transformations in $T_{E}(X)$ forms a subsemigroup of $T_E(X)$ denoted by \[ {\cal O}_{E}(X)=\{f\in T_{E}(X)\colon \ \forall\, x, y\in X, \ x\leq y \mbox{ implies } f(x)\leq f(y)\}. \] The nature of regular elements in ${\cal O}_{E}(X)$ is described and the Green's equivalences on ${\cal O}_{E}(X)$ are characterized completely.

Keywords

Equivalence Relation Regular Element Order Preserve Consecutive Point Preserve Transformation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Department of Mathematics, Xinyang Normal University, Xinyang, Henan 464000P. R. China
  2. 2.Department of Information Science, Jiangsu Polytechnic University Changzhou, Jiangsu 213000P. R. China

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