降序变换半群<inline-formula><span class="inline-formula-span"> \Large S_n^-</span><img text_id='Z-20190802021717-3' class='formula-img' style='display:none;' src='yndz-20180143-Y_Z-20190802021717-3.jpg'><alternatives></alternatives></inline-formula>上的自然偏序关系

降序变换半群 \Large S_n^-上的自然偏序关系

摘要: 设 T_X 是全序集 X = \left\ 1 <2 < \cdots < n \right\ 上的全变换半群，则 S_n^ - = \ f \in T_X:\forall x \in X,f(x) \leqslant x\ 是 T_X 的降序变换子半群. 赋予降序变换半群 S_n^ - 自然偏序关系，给出了 S_n^ - 的特征，刻画了 S_n^ - 的相容元，描述了 S_n^ - 的极小元和极大元.

Abstract: Let T_X be the full transformation semigroup on a total order set X = \ 1 <2 < \cdots < n\ . Then S_n^ - = \left\ f \in T_X:\forall x \in X,f(x) \le x \right\ is a subsemigroup of T_X. We endow the order-decreasing transformation semigroup S_n^ - with the natural partial order. With respect to this partial order, we investigate when two elements of \Large S_n^- are related, then find elements of \Large S_n^- which are compatible with the order. Also, we characterize the minimal elements and the maximal elements of \Large S_n^- .