正整数幂的主滤子的序同构幺半群
The monoid of order isomorphisms of principal filters of a power of the positive integers
摘要：
设$n$是任意正整数，$\mathscr{I\!\!P\!F}(\mathbb{N}^n)$是正整数集合$\mathbb{N}$的$n$次幂的主滤子与乘积阶之间的所有序同构的半群.研究了半群$\mathscr{I\!\!P\!F}(\mathbb{N}^n)$的代数性质。特别地，我们证明了$\mathscr{I\!\!P\!F}(\mathbb{N}^n)$是一个双单的$E$ -酉的$F$ -逆半群，刻画了$\mathscr{I\!\!P\!F}(\mathbb{N}^n)$及其极大子群上的Green关系.证明了半群$\mathscr{I\!\!P\!F}(\mathbb{N}^n)$同构于双圈幺半群${\mathscr{C}}^n(p,q)$的直$n$次幂与置换群$\mathscr{S}_n$的半直积.证明了半群$\mathscr{I\!\!P\!F}(\mathbb{N}^n)$上的所有非恒等同余$\mathfrak{C}$都是群，并刻画了$\mathscr{I\!\!P\!F}(\mathbb{N}^n)$上的最小群同余。证明了$\mathscr{I\!\!P\!F}(\mathbb{N}^n)$上的Hausdorff移位连续拓扑都是离散的，并讨论了半群$\mathscr{I\!\!P\!F}(\mathbb{N}^n)$嵌入到类紧拓扑半群中.
英文摘要：
Let $n$ be any positive integer and $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$ be the semigroup of all order isomorphisms between principal filters of the $n$-th power of the set of positive integers $\mathbb{N}$ with the product order. We study algebraic properties of the semigroup $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$. In particular, we show that $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$ is a bisimple, $E$-unitary, $F$-inverse semigroup, describe Green's relations on $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$ and its maximal subgroups. We show that the semigroup $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$ is isomorphic to the semidirect product of the direct $n$-th power of the bicyclic monoid ${\mathscr{C}}^n(p,q)$ by the group of permutation $\mathscr{S}_n$. Also we prove that every non-identity congruence $\mathfrak{C}$ on the semigroup $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$ is group and describe the least group congruence on $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$. We show that every Hausdorff shift-continuous topology on $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$ is discrete and discuss embedding of the semigroup $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$ into compact-like topological semigroups.
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