Narrow Results

Let $X$ be a nonempty set and let ${𝒫}{𝒯}(X)$ be the monoid of all partial transformations of $X$. Given a nonempty subset $Y$ of $X$, we denote by ${𝒫}{𝒯}(X,Y)$ the subsemigroup of ${𝒫}{𝒯}(X)$ of all partial transformations with range contained in $Y$. In 2014, Fernandes and Sanwong [On the ranks of semigroups of transformations on a finite set with restricted range, Algebra Colloquium. 21(3) (2014) 497–510] proved that the semigroup ${𝒫}{ℱ}(X,Y)=\{\alpha \in {𝒫}{𝒯}(X,Y):X\alpha =Y\alpha \}$ is the largest regular subsemigroup of ${𝒫}{𝒯}(X,Y)$ and showed that this subsemigroup determines Green’s relations on ${𝒫}{𝒯}(X,Y)$. In this paper, we classify completely the maximal subsemigroups of ${𝒫}{ℱ}(X,Y)$ when $\left|Y\right|=r$. Moreover, we prove that the rank of ${𝒫}{ℱ}(X,Y)$ is equal to ${(r+1)}^{n-r}+2$ when $\left|X\right|=n$ and $\left|Y\right|=r$.

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 ${{𝒪}{ℰ}}_{n,k}$.

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 $\{1,\dots ,n\}$. 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.