论文标题
poset $（\ mathbb {n}^{3}，\ leqslant）$的单调注射式部分自我图的单调局部自我图，带有cofinite域和图像
The monoid of monotone injective partial selfmaps of the poset $(\mathbb{N}^{3},\leqslant)$ with cofinite domains and images
论文作者
论文摘要
令$ n $为正整数$ \ geqslant 2 $和$ \ mathbb {n}^n _ {\ leqslant} $是$ n $ - 正整数的$ n $ the，并具有$ \ mathbb {n} $的常规订单订单。在论文中，我们研究了$ \ mathbb {n}^n _ {\ leqslant} $的注入性部分单调自我图的半群，并使用Cofinite域和图像。我们表明，半群$ \ Mathscr {p \！o} \！_ {\ infty}（\ MathBB {n}^n _ {\ leqslant}）$ isomorphic是组$ \ math $ scr y Math， $ n $ element集，并描述$ \ mathscr {p \！o} \！_ {\ infty}（\ Mathbb {n}^n _ {\ leqslant}）$的$ \ mathscr {p \！o} \！_ {\ mathbb {\ mathbb {同样在这种情况下，在$ n = 3 $的情况下，我们描述了半群$ \ mathscr {p \！在Semigroup $ \ Mathscr {p \！o} \！_ {\ infty}（\ mathbb {n}^3 _ {\ leqslant}）$。特别是我们表明$ \ Mathscr {d} = \ Mathscr {J} $ in $ \ MathScr {p \！
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 $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^3_{\leqslant})$.
