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Full-Text Articles in Physical Sciences and Mathematics
On The Monoid Of Partial Isometries Of A Cycle Graph, Vitor H. Fernandes, Tania Paulista
On The Monoid Of Partial Isometries Of A Cycle Graph, Vitor H. Fernandes, Tania Paulista
Turkish Journal of Mathematics
In this paper we consider the monoid $DPC_n$ of all partial isometries of an $n$-cycle graph $C_n$. We show that $DPC_n$ is the submonoid of the monoid of all oriented partial permutations on an $n$-chain whose elements are precisely all restrictions of a dihedral group of order $2n$. Our main aim is to exhibit a presentation of $DPC_n$. We also describe Green's relations of $DPC_n$ and calculate its cardinality and rank.
On The Rank Of Generalized Order-Preserving Transformation Semigroups, Haytham Darweesh Mustafa Abusarri̇s, Gonca Ayik
On The Rank Of Generalized Order-Preserving Transformation Semigroups, Haytham Darweesh Mustafa Abusarri̇s, Gonca Ayik
Turkish Journal of Mathematics
For any two non-empty (disjoint) chains $X$ and $Y$, and for a fixed order-preserving transformation $\theta : Y \rightarrow X$, let $\mathcal{GO} (X,Y; \theta )$ be the generalized order-preserving transformation semigroup. Let $\mathcal{O}(Z)$ be the order-preserving transformation semigroup on the set $Z=X\cup Y$ with a defined order. In this paper, we show that $\mathcal{GO}(X,Y;\theta)$ can be embedded in $O(Z,Y)=\{\, \alpha\in \mathcal{O}(Z)\, :\, Z\alpha \subseteq Y\,\}$, the semigroup of order-preserving transformations with restricted range. If $\theta \in \mathcal{GO}(Y,X)$ is one-to-one, then we show that $\mathcal{GO}(X,Y; \theta)$ and $O(X, im (\theta))$ are isomorphic semigroups. If we suppose that $\left X \right =m$,\, …