Quasi-idempotents in finite semigroup of full order-preserving transformations

Abstract

Let \(X_n\) be the finite set \(\left\lbrace1,2,3\cdots,n\right\rbrace\) and \(\mathcal{O}_n\) defined by \(O_n = \lbrace \alpha\in T_n\colon (\forall x,y \in X_n),\; x\leq y\rightarrow x\alpha \leq y\alpha\rbrace\) be the semigroup of full order-preserving mapping on \(X_n\). A~transformation \(\alpha\) in \(\mathcal{O}_n\) is called quasi-idempotent if \(\alpha\neq \alpha^2= \alpha^4\). We characterise quasi-idempotent in \(\mathcal{O}_n\) and show that the semigroup \(\mathcal{O}_n\) is quasi-idempotent generated. Moreover, we obtained an upper bound for quasi-idempotents rank of \(\mathcal{O}_n\), that is, we showed that the cardinality of a minimum quasi-idempotents generating set for \(\mathcal{O}_n\) is less than or equal to \(\lceil \frac{3(n-2)}{2}\rceil\) where \(\lceil x\rceil\) denotes the least positive integer \(m\) such that \(x \leq m < x + 1\).