摘要
文中定义了半格上的算子,给出了半格上算子的几个等价描述,得到如下定理:设(L,∨)表示L是一并半格,F是L到自身的一个映射,则如下几条等价:(1)F是L上的闭包算子;(2)x,y∈L,x∨F(F(x)∨F(y))=F(x∨y);(3)F是L上的闭包算子,且满足F(F(x)∨F(y))=F(x∨y);(4)F满足:x≤F(x)且F(F(x)∨F(y))=F(x∨y).另外。
The operator on semilattice was defined and the following theroem was obtained:Let (L,∨) and F be a join semilattice and an operator from L to L litself respectively,then the following conditions are equavalent: (1) F is a closure operator on L ; (2) x,y∈L,x∨F(F(x)∨F(y))=F(x∨y); (3) F is a closure operator on L ,moreover,it is satisfied with F(F(x)∨F(y))=F(x∨y); (4) F is satisfied with x≤F(x) and F(F(x)∨F(y))=F(x∨y). Moreover,a map from Su(x) ,which is the power set of a set X ,to Su(X) itself is a topological interior operator iff the following equation: X-A∩I(A)∩I 2(B)=I(X)-I(A∩B) is satisfied.
关键词
并半格
闭包算子
拓扑内部算子
join semilattice
closure operator
topological interior operator