降序变换半群 \Large S_n^- 上的自然偏序关系
A natural partial order on the semigroups \Large S_n^- of order-decreasing transformations
-
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: LetT_X be the full transformation semigroup on a total order setX = \ 1 <2 < \cdots < n\ . ThenS_n^ - = \left\ f \in T_X:\forall x \in X,f(x) \le x \right\ is a subsemigroup ofT_X . We endow the order-decreasing transformation semigroupS_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^- .