Özet:

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 $\left X \right =m$,\, $\left Y\right =n$, and $\left im(\theta) \right =r$ where $m,n,r\in \mathbb{N}$, then we find the rank of $\mathcal{GO}(X,Y;\theta )$ when $\theta $ is one-to-one but not onto. Moreover, we find lower bounds for $rank (\mathcal{GO}(X,Y;\theta ))$ when $\theta $ is neither one-to-one nor onto and when $\theta $ is onto but not one-to-one.

Anahtar Kelimeler: