论文标题
格林的关系和针对人物是徒的转换的单位型号
Green's relations and unit-regularity for semigroup of transformations whose characters are bijective
论文作者
论文摘要
令$ x $为非发行集，$ \ Mathcal {p} = \ {x_i \ colon i \ in I \} $是$ x $的分区。用$ t（x，\ mathcal {p}）$表示$ x $的所有转换的半群，保留$ \ nathcal {p} $。在本文中，我们研究了所有转换的Semigroup $ \ Mathcal {b}（x，x，x，\ mathcal {p}）$ $ f \ in t（x，x，\ nathcal {p} $ $χ^{（f）} \ colon i \ to i $是$iχ^{（f）} = j $定义的$ f $的字符（map），每当$ x_if \ subseteq x_j $时。我们在$ \ Mathcal {b}（x，x，\ Mathcal {p}）$中描述单位定型元素，并确定$ \ MATHCAL {B}（x，x，x，\ MATHCAL {p}）$何时是单位规范的SemoGroup。或者，我们可以证明$ \ Mathcal {B}（x，\ Mathcal {p}）$是常规的半群。我们描述了格林在$ \ Mathcal {b}（x，x，\ Mathcal {p}）$上的关系，并证明$ \ Mathcal {d} = \ Mathcal {J} $ on $ \ MATHCAL {b} $我们还为$ \ Mathcal {d} = \ Mathcal {J} $提供了必要且充分的条件，$ \ Mathcal {b}（x，x，x，\ Mathcal {p}）$。我们以猜想结束了论文。
Let $X$ be a nonempty set and $\mathcal{P}=\{X_i\colon i\in I\}$ be a partition of $X$. Denote by $T(X, \mathcal{P})$ the semigroup of all transformations of $X$ that preserve $\mathcal{P}$. In this paper, we study the semigroup $\mathcal{B}(X,\mathcal{P})$ of all transformations $f\in T(X, \mathcal{P})$ such that $χ^{(f)}\in {\rm Sym}(I)$, where ${\rm Sym}(I)$ is the symmetric group on $I$ and $χ^{(f)}\colon I \to I$ is the character (map) of $f$ defined by $iχ^{(f)}=j$ whenever $X_if\subseteq X_j$. We describe unit-regular elements in $\mathcal{B}(X,\mathcal{P})$, and determine when $\mathcal{B}(X,\mathcal{P})$ is a unit-regular semigroup. We alternatively prove that $\mathcal{B}(X,\mathcal{P})$ is a regular semigroup. We describe Green's relations on $\mathcal{B}(X,\mathcal{P})$, and prove that $\mathcal{D} = \mathcal{J}$ on $\mathcal{B}(X,\mathcal{P})$ when $\mathcal{P}$ is finite. We also give a necessary and sufficient condition for $\mathcal{D} = \mathcal{J}$ on $\mathcal{B}(X,\mathcal{P})$. We end the paper with a conjecture.
