论文标题
定期，单位定型和愿意的转换分区的元素
Regular, Unit-regular, and Idempotent elements of semigroups of transformations that preserve a partition
论文作者
论文摘要
令$ x $为套装，$ \ nathcal {t} _x $是$ x $上的完整转换半群。对于$ x $的分区$ \ MATHCAL {P} $，我们考虑semigroups $ t（x，x，\ mathcal {p}）= \ {f \ in \ Mathcal {t} _x | （\ forall x_i \ in \ Mathcal {p}）（\存在x_j \ in \ Mathcal {p}）\; x_i f \ subseteq x_j \} $，$σ（x，x，x，x，\ nathcal {p}） \ in \ Mathcal {p}）\; xf \ cap x_i \ neq \ emptySet \} $和$γ（x，x，\ mathcal {p}）= \ {f \ in \ Mathcal {t} _x |（\ forall x_i \ in \ nathcal in \ Mathcal {p}）（p}）（p}） x_i f = x_j \} $。我们表征了有限$ x $的$ t（x，\ mathcal {p}）$和$σ（x，x，\ mathcal {p}）$的单位定型元素。我们讨论了$γ（x，\ mathcal {p}）$与保留$ \ Mathcal {p} $的某些转换的某些半群之间的集合。我们表征并计算$γ（x，\ Mathcal {p}）$的常规元素和识别元素。对于有限的$ x $，我们证明$γ（x，\ mathcal {p}）$的每个常规元素都是单位定型的，并且还计算$γ（x，\ mathcal {p}）$的大小。
Let $X$ be a set and $\mathcal{T}_X$ be the full transformation semigroup on $X$. For a partition $\mathcal{P}$ of $X$, we consider semigroups $T(X, \mathcal{P}) = \{f\in \mathcal{T}_X| (\forall X_i\in \mathcal{P}) (\exists X_j \in \mathcal{P})\;X_i f \subseteq X_j\}$, $Σ(X, \mathcal{P}) = \{f\in T(X, \mathcal{P})|(\forall X_i \in \mathcal{P})\; Xf \cap X_i \neq \emptyset\}$, and $Γ(X, \mathcal{P}) = \{f\in \mathcal{T}_X|(\forall X_i\in \mathcal{P})(\exists X_j\in \mathcal{P})\; X_i f = X_j\}$. We characterize unit-regular elements of both $T(X, \mathcal{P})$ and $Σ(X, \mathcal{P})$ for finite $X$. We discuss set inclusion between $Γ(X, \mathcal{P})$ and certain semigroups of transformations preserving $\mathcal{P}$. We characterize and count regular elements and idempotents of $Γ(X, \mathcal{P})$. For finite $X$, we prove that every regular element of $Γ(X, \mathcal{P})$ is unit-regular and also calculate the size of $Γ(X, \mathcal{P})$.
