论文标题
在有限冠上所有部分自动形态的反向半群的等级
The rank of the inverse semigroup of all partial automorphisms on a finite crown
论文作者
论文摘要
对于$ n \ in \ mathbb {n} $，让$ [n] = \ {1，2，\ ldots，n \} $为$ n $ - 元素集。像往常一样，我们用$ i_n $表示$ [n] $的对称反向半群，即，在映射的组成下，$ [n] $的部分一对一转换semigroup。 The crown (cycle) $\cal{C}_n$ is an $n$-ordered set with the partial order $\prec$ on $[n]$, where the only comparabilities are $$1 \prec 2 \succ 3 \prec 4 \succ \cdots \prec n \succ 1 ~~\mbox{ or }~~ 1 \succ 2 \prec 3 \succ 4 \prec \cdots \succ n \ prec 1。$$我们说，如果$ x \ prec y $ $ x \ prec y $表示$xα\ precyα$，则$xα\ precyα$，对于所有$ x，y $，y_n $中的转换$α\是订单提供的。在本文中，我们研究了有限的皇冠$ \ cal {c} _n $上所有部分汽车的反向半群$ ic_n $。我们考虑元素，确定一组最小尺寸的生成集，并计算$ ic_n $的排名。
For $n \in \mathbb{N}$, let $[n] = \{1, 2, \ldots, n\}$ be an $n$ - element set. As usual, we denote by $I_n$ the symmetric inverse semigroup on $[n]$, i.e. the partial one-to-one transformation semigroup on $[n]$ under composition of mappings. The crown (cycle) $\cal{C}_n$ is an $n$-ordered set with the partial order $\prec$ on $[n]$, where the only comparabilities are $$1 \prec 2 \succ 3 \prec 4 \succ \cdots \prec n \succ 1 ~~\mbox{ or }~~ 1 \succ 2 \prec 3 \succ 4 \prec \cdots \succ n \prec 1.$$ We say that a transformation $α\in I_n$ is order-preserving if $x \prec y$ implies that $xα\prec yα$, for all $x, y $ from the domain of $α$. In this paper, we study the inverse semigroup $IC_n$ of all partial automorphisms on a finite crown $\cal{C}_n$. We consider the elements, determine a generating set of minimal size and calculate the rank of $IC_n$.
