Narrow Results

Let $1\le k\le n$. We denote by ${{𝒪}{ℰ}}_{n,k}$ the monoid of all order-preserving k-extensive full transformations on $\{1,\dots ,n\}$ ordered in the standard way. In this paper, it is shown that ${{𝒪}{ℰ}}_{n,k}$ is not a regular semigroup in general, but it is an abundant semigroup. Moreover, we prove that ${{𝒪}{ℰ}}_{n,k}$ is idempotent-generated and compute the idempotent rank and rank of the monoid ${{𝒪}{ℰ}}_{n,k}$. Moreover, we determine the maximal idempotent generated subsemigroups of the monoid ${{𝒪}{ℰ}}_{n,k}$.

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.

It is known that the ranks of the semigroups 𝒮ing_n and 𝒮𝒫𝒯_n (the semigroups of singular self-maps and partial transformations on X_n = {1, 2, …, n}, respectively) are , respectively. The idempotent ranks, defined as the smallest number of idempotents generating set, of 𝒮ing_n and 𝒮𝒫𝒯_n are the same value as the ranks, respectively. Idempotent can be seen as a special case (with m = r = 1) of (m, r)-potent. In this paper, we determine the (m, r)-potent ranks, defined as the smallest number of (m, r)-potent generating set, of the semigroups 𝒮ing_n and 𝒮𝒫𝒯_n.

Let ${{𝒪}}_n$ be the semigroup of all singular order-preserving mappings on the finite set $X_n=\{1,2,\dots ,n\}$. It is known that ${{𝒪}}_n$ is generated by its set of idempotents of rank $n-1$, and its rank and idempotent rank are $n$ and $2n-2$, respectively. In this paper, we study the structure of the semigroup generated by any nonempty subset of idempotents of rank $n-1$ in ${{𝒪}}_n$. We also calculate its rank and idempotent rank.

Let ${{𝒫}}_n$ and ${{𝒮}{𝒫}}_n$ be the partial transformation and the strictly partial transformation semigroups on the finite set $X_n=\{1,2,\dots ,n\}$. It is well known that the ranks of the semigroups ${𝒫}(n,r)=\{\alpha \in {{𝒫}}_n:|\mathrm{im}(\alpha )|\le r\}$ and ${𝒮}{𝒫}(n,r)=\{\alpha \in {{𝒮}{𝒫}}_n:|\mathrm{im}(\alpha )|\le r\}$ are $S(n+1,r+1)$, for $2\le r\le n-1$, and $(r+1)S(n,r+1)$, for $2\le r\le n-2$, respectively. The idempotent rank, defined as the smallest number of idempotents generating set, of the semigroup ${𝒫}(n,r)$ has the same value as the rank. Idempotent can be seen as a special case (with $p=q=1$) of $(p,q)$-potent. In this paper, we determine the $(p,q)$-potent ranks, defined as the smallest number of $(p,q)$-potents generating set, of the semigroups ${𝒫}(n,r)$, for $2\le r\le n-1$, and ${𝒮}{𝒫}(n,r)$, for $2\le r\le n-2$.

For $n\in \mathbb{N}$, let ${𝒮}in{g}_n$ be the semigroup of all singular mappings on $X_n=\{1,2,\dots ,n\}$. For each nonempty subset $A$ of $X_n$, let ${{𝒮}}_n(A)=\{\alpha \in {𝒮}in{g}_n:(\forall k\in A)x\le k\Rightarrow x\alpha \le k\}$ be the semigroup of all $A$-decreasing mappings on $X_n$. In this paper we determine the rank and idempotent rank of the semigroup ${{𝒮}}_n(A)$.

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.

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.