Volume 36 • Number 1 • 2013   • Combinatorial Results for Certain Semigroups of Transformations Preserving Orientation and a Uniform Partition Lei Sun Abstract. 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. 2010 Mathematics Subject Classification: 20M20, 05A10 Full text: PDF