Combinatorial Results for Certain Semigroups of Transformations Preserving Orientation and a Uniform Partition

Lei Sun

Bulletin of the Malaysian Mathematical Society, Tome 36 (2013) no. 1

Résumé

Let ${\mathcal T}_X$ be the full transformation semigroup on a set $X$ and $E$ be a non-trivial equivalence on $X$. The set

$T_E (X) =\{ f\in {\mathcal T}_X : \forall \, (x,y)\in E,\, (f(x),f(y))\in E \}$

is a subsemigroup of ${\mathcal T}_ X $. For a finite totally ordered set $X$ and a convex equivalence $E$ on $X$, the set of all the orientation-preserving transformations in $T_E(X)$ forms a subsemigroup of $T_E(X)$ denoted by $OP_E(X)$. In this paper, under the hypothesis that the totally ordered set $X$ is of cardinality $mn\,\,(m,n\geq 2)$ and the equivalence $E$ has $m$ classes such that each $E$-class contains $n$ consecutive points, we calculate the cardinality of the semigroup $OP_E(X)$, and that of its idempotents.

Détail

Classification : 20M20, 05A10