The decreasing and monotone injective partial monoid on a finite chain

Abstract

In this paper, we consider the monoid \(\mathcal{DORI}_{n}\) consisting of all monotone and order-decreasing partial injective transformations, \(I(n, p) = \{ \alpha \in \mathcal{DORI}_{n} : |\) Im \(\alpha| \leq p \}\) the two-sided ideal of \(\mathcal{DORI}_{n}\), and \({RQ}_{p}(n)\) the Rees quotient of \(I(n, p)\) on a chain with \(n\) elements. We calculate the cardinality of \(\mathcal{DORI}_{n}\), characterize the Green's relations and their starred analogue for any structure \(S\in\{\mathcal{DORI}_{n}, I(n, p), {RQ}_{p}(n)\}\). We demonstrate that for any structure \(S\) among \(\{\mathcal{DORI}_{n}, I(n, p), {RQ}_{p}(n)\}\), the structure is abundant for all values of \(n\); specifically, \(\mathcal{DORI}_{n}\) is shown to be an ample monoid, and compute the rank of the Rees quotient \({RQ}_{p}(n)\) and the two-sided ideal \(I(n, p)\); as a special case, we obtain the rank of the monoid \(\mathcal{DORI}_{n}\) to be \(3n - 2\). Finally, we characterize all the maximal subsemigroups of the structure \(S\) among \(\{\mathcal{DORI}_{n}, I(n, p), {RQ}_{p}(n)\}\).