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Let ${{𝒪}{𝒫}{ℰ}}_n$ be the monoid of all orientation-preserving and extensive full transformations on $\{1,2,\dots ,n\}$ ordered in the standard way. In this paper, we determine the minimum generating set and the minimum idempotent generating set of ${{𝒪}{𝒫}{ℰ}}_n$, and so the rank and the idempotent rank of ${{𝒪}{𝒫}{ℰ}}_n$ are obtained. Moreover, we describe maximal subsemigroups and maximal idempotent generated subsemigroups of ${{𝒪}{𝒫}{ℰ}}_n$ and completely obtain their classifications.

The variant of a semigroup $S$ with respect to an element $a\in S$, denoted $S^a$, is the semigroup with underlying set $S$ and operation ⋆ defined by $x\star y=xay$ for $x,y\in S$. In this paper, we study variants ${{𝒯}}_X^a$ of the full transformation semigroup ${{𝒯}}_X$ on a finite set $X$. We explore the structure of ${{𝒯}}_X^a$ as well as its subsemigroups $\mathrm{\text{Reg}}({{𝒯}}_X^a)$ (consisting of all regular elements) and ${{ℰ}}_X^a$ (consisting of all products of idempotents), and the ideals of $\mathrm{\text{Reg}}({{𝒯}}_X^a)$. Among other results, we calculate the rank and idempotent rank (if applicable) of each semigroup, and (where possible) the number of (idempotent) generating sets of the minimal possible size.

We give a thorough structural analysis of the principal one-sided ideals of arbitrary semigroups, and then apply this to full transformation semigroups and symmetric inverse monoids. One-sided ideals of these semigroups naturally occur as semigroups of transformations with restricted range or kernel.

Let ${{𝒪}{𝒫}{ℰ}}_n$ be the monoid of all orientation-preserving and extensive full transformations on $\{1,\dots ,n\}$. In this paper, we compute the rank and the idempotent rank of the ideals of the monoid ${{𝒪}{𝒫}{ℰ}}_n$. Moreover, we determine the maximal subsemigroups as well as the maximal subsemibands of the ideals of the monoid ${{𝒪}{𝒫}{ℰ}}_n$. Our work extends previous results found in the literature.

For n ∈ ℕ, let O_n be the semigroup of all singular order-preserving mappings on [n]= {1,2,…,n}. For each nonempty subset A of [n], let O_n(A) = {α ∈ O_n: (∀ k ∈A) kα ≤ k} be the semigroup of all order-preserving and A-decreasing mappings on [n]. In this paper it is shown that O_n(A) is an abundant semigroup with n-1𝒟*-classes. Moreover, O_n(A) is idempotent-generated and its idempotent rank is 2n-2- |A\ {n}|. Further, it is shown that the rank of O_n(A) is equal to n-1 if 1 ∈ A, and it is equal to n otherwise.