偏序集$(\mathbb{N}^{3},\leqslant)$的单调内射部分自映射幺半群
The monoid of monotone injective partial selfmaps of the poset $(\mathbb{N}^{3},\leqslant)$ with cofinite domains and images
摘要：
设$n$为正整数$\geqslant 2$，$\mathbb{N}^n_{\leqslant}$为正整数的$n$次幂，其乘积阶为$\mathbb{N}$上的通常阶.本文研究了$\mathbb{N}^n_{\leqslant}$的部分单调内射自映射半群的上有限域和象.证明了半群$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^n_{\leqslant})$的单位群$H(\mathbb{I})$同构于$n$ -元集的置换群$\mathscr{S}_n$，并刻画了$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^n_{\leqslant})$的幂等元子半群.同样在$n=3$的情况下，我们将半群$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^3_{\leqslant})$的元素的性质描述为偏序集$\mathbb{N}^3_{\leqslant}$的部分双射和半群$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^3_{\leqslant})$上的格林关系。特别是，我们表明，$\mathscr{D}=\mathscr{J}$在$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^3_{\leqslant})$。
英文摘要：
Let $n$ be a positive integer $\geqslant 2$ and $\mathbb{N}^n_{\leqslant}$ be the $n$-th power of positive integers with the product order of the usual order on $\mathbb{N}$. In the paper we study the semigroup of injective partial monotone selfmaps of $\mathbb{N}^n_{\leqslant}$ with cofinite domains and images. We show that the group of units $H(\mathbb{I})$ of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^n_{\leqslant})$ is isomorphic to the group $\mathscr{S}_n$ of permutations of an $n$-element set, and describe the subsemigroup of idempotents of $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^n_{\leqslant})$. Also in the case $n=3$ we describe the property of elements of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^3_{\leqslant})$ as partial bijections of the poset $\mathbb{N}^3_{\leqslant}$ and Green's relations on the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^3_{\leqslant})$. In particular we show that $\mathscr{D}=\mathscr{J}$ in $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^3_{\leqslant})$.
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