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

摘要

Let ${cal T}_X$ be the full transformation semigroup on the set $X$, [ T_{E}(X)={fin {cal T}_Xcolon 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 $mgeq 2$ and $ngeq 3$. 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)={fin T_{E}(X)colon forall, x, yin X, xleq 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.