PRODUCT OF SEMI-TRANSPOSITIONS IN FINITE SEMIGROUP OF INJECTIVE ORDER-PRESERVING TRANSFORMATIONS

PRODUCT OF SEMI-TRANSPOSITIONS IN FINITE SEMIGROUP

Abstract

 Let $X_n$ be the finite totally ordered set $\{1,2,\cdots,n\}$, $\mathcal{IO}_n$ be the semigroup of all injective order-preserving transformation of $X_n$ and $\mathcal{IO}_{n,r}=\{\alpha\in \mathcal{IO}_n: |im(\alpha)|\leq r\}$ for $(1\leq r\leq n-1)$ be the ideals of injective order-preserving transformations on $X_n$. The semigroup $\mathcal{IO}_{n}$ on $X_n$ is an inverse semigroup and so cannot be generated by its idempotents. In a search for generating set for $\mathcal{IO}_n$ in this article, we identify a class of quasi-idempotents (i.e elements $\alpha$ in $\mathcal{IO}_{n}$ satisfying $\alpha\neq \alpha^2=\alpha^4.$) which we refer to as semi-transpositions and showed that the ideals $\mathcal{IO}_{n,r}$ are generated by semi-trnaspositions. The semi-transposition rank of $\mathcal{IO}_{n,r}$ (defined to be the minimum of such generating set) is obtained to be $2\binom{n}{r}-2.$