On certain subsemigroups of finite oriented and order-decreasing full transformations

TL;DR

This paper investigates the algebraic structure of a specific semigroup of transformations on finite chains, determining its size, nilpotent elements, minimal generators, rank, and maximal subsemigroups.

Contribution

It provides explicit formulas and characterizations for the semigroup RD(n,r), including its cardinality, nilpotent elements, minimal generating set, rank, and maximal subsemigroups.

Findings

Cardinality of RD(n,r) determined

Number of nilpotent elements characterized

Maximal subsemigroups characterized

Abstract

Let $\mathcal{ORD}_{n}$ be the semigroup consisting of all oriented and order-decreasing full transformations on the finite chain $X_{n}=\{ 1<\cdots<n \}$, and for $1\leq r\leq n-1$, let $$\mathcal{ORD}(n,r) =\{\alpha \in \mathcal{ORD}_{n}\, :\, \lvert \textrm{im}(\alpha )\rvert \leq r\}.$$ In this paper, we determine the cardinality of $\mathcal{ORD}(n,r)$ and the number of nilpotent elements of $\mathcal{ORD}(n,r)$, we find a minimal generating set and the rank of $\mathcal{ORD}(n,r)$, and moreover, we characterize all maximal subsemigroups of $\mathcal{ORD}(n,r)$ for each $3\leq r\leq n-1$.