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Submitted by Isabel Melo (imelo@sa.isel.pt) on 2020-05-13T18:05:16Z No. of bitstreams: 1 Ranks_TMQuinteiro.pdf: 404654 bytes, checksum: db135bf332f766653a2b0e9c4e96ef20 (MD5) Made available in DSpace on 2020-05-13T18:05:16Z (GMT). No. of bitstreams: 1 Ranks_TMQuinteiro.pdf: 404654 bytes, checksum: db135bf332f766653a2b0e9c4e96ef20 (MD5) Previous issue date: 2019-04-19 info:eu-repo/semantics/publishedVersion

In this paper, we study partial automorphisms and, more generally, injective partial endomorphisms of a finite undirected path from Semigroup Theory perspective. Our main objective is to give formulas for the ranks of the monoids $$\mathrm {IEnd}(P_n)$$ and $$\mathrm {PAut}(P_n)$$ of all injective partial endomorphisms and of all partial automorphisms of the undirected path $$P_n$$ with n vertices. We also describe Green’s relations of $$\mathrm {PAut}(P_n)$$ and $$\mathrm {IEnd}(P_n)$$ and calculate their cardinals.

In this note we consider the monoid $\mathcal{PODI}_n$ of all monotone partial permutations on $\{1,\ldots,n\}$ and its submonoids $\mathcal{DP}_n$, $\mathcal{POI}_n$ and $\mathcal{ODP}_n$ of all partial isometries, of all order-preserving partial permutations and of all order-preserving partial isometries, respectively. We prove that both the monoids $\mathcal{POI}_n$ and $\mathcal{ODP}_n$ are quotients of bilateral semidirect products of two of their remarkable submonoids, namely of extensive and of co-extensive transformations. Moreover, we show that $\mathcal{PODI}_n$ is a quotient of a semidirect product of $\mathcal{POI}_n$ and the group $\mathcal{C}_2$ of order two and, analogously, $\mathcal{DP}_n$ is a quotient of a semidirect product of $\mathcal{ODP}_n$ and $\mathcal{C}_2$., Comment: 10 pages, submitted

Let X n be a chain with n elements (n ∈ ℕ), and let 𝒪𝒫 n be the monoid of all orientation-preserving transformations of X n . In this article, for any nonempty subset Y of X n , we consider the subsemigroup 𝒪𝒫 n (Y) of 𝒪𝒫 n of all transformations with range contained in Y: We describe the largest regular subsemigroup of 𝒪𝒫 n (Y), which actually coincides with its subset of all regular elements. Also, we determine when two semigroups of the type 𝒪𝒫 n (Y) are isomorphic and calculate their ranks. Moreover, a parallel study is presented for the correspondent subsemigroups of the monoid 𝒪ℛ n of all either orientation-preserving or orientation-reversing transformations of X n .

In this article we consider the monoid 𝒪 m × n of all order-preserving full transformations on a chain with mn elements that preserve a uniform m-partition and its submonoids and of all extensive transformations and of all co-extensive transformations, respectively. We determine their ranks and construct a bilateral semidirect product decomposition of 𝒪 m × n in terms of and .

In this paper we consider the monoid $\mathcal {OR}_{n}$ of all full transformations on a chain with n elements that preserve or reverse the orientation, as well as its submonoids $\mathcal {OD}_{n}$ of all order-preserving or order-reversing elements, $\mathcal {OP}_{n}$ of all orientation-preserving elements and $\mathcal {O}_{n}$ of all order-preserving elements. By making use of some well known presentations, we show that each of these four monoids is a quotient of a bilateral semidirect product of two of its remarkable submonoids.

In this paper we give presentations for the monoid $${\mathcal {DP}}_n$$ of all partial isometries on $$\{1,\ldots ,n\}$$ and for its submonoid $${\mathcal {ODP}}_n$$ of all order-preserving partial isometries.

Let $X$ be a finite or infinite chain and let $O(X)$ be the monoid of all endomorphisms of $X$. In this paper, we describe the largest regular subsemigroup of $O(X)$ and Green's relations on $O(X)$. In fact, more generally, if $Y$ is a nonempty subset of $X$ and $O(X,Y)$ the subsemigroup of $O(X)$ of all elements with range contained in $Y$, we characterize the largest regular subsemigroup of $O(X,Y)$ and Green's relations on $O(X,Y)$. Moreover, for finite chains, we determine when two semigroups of the type $O(X,Y)$ are isomorphic and calculate their ranks., To appear in Semigroup Forum

The rank of a semigroup, an important and relevant concept in Semigroup Theory, is the cardinality of a least-size generating set. Semigroups of transformations that preserve or reverse the order or the orientation as well as semigroups of transformations preserving an equivalence relation have been widely studied over the past decades by many authors. The purpose of this article is to compute the ranks of the monoid 𝒪ℛ m×n of all orientation-preserving or orientation-reversing full transformations on a chain with mn elements that preserve a uniform m-partition and of its submonoids 𝒪𝒫 m×n of all orientation-preserving transformations and 𝒪𝒟 m×n of all order-preserving or order-reversing full transformations. These three monoids are natural extensions of 𝒪 m×n , the monoid of all order-preserving full transformations on a chain with mn elements that preserve a uniform m-partition. Given m, n ≥ 2, we show that the rank of 𝒪𝒫 m×n is equal to , for m > 2, and equal to , for m = 2, the rank of 𝒪𝒟 m×n is equal...