分配半格的偏序集表示
Poset representations of distributive semilattices
摘要：
本文证明了对任意分配并半格S，存在一个交半格P，它的值为零，且映射f：PxP-->S使得f（x，z）≤ f（x，y）vf（y，z）且x ≤ y蕴涵f（x，y）=0，对P中的所有x，y，z，满足下列条件：（i）f（y，x）=0意味着x=y，对于P中的所有x<=y。（ii）对于P中的所有x \leq y和S中的所有a，b，如果f（y，x）=avb，则有一个正整数n和一个分解x=x_0<=x_1<=. <=x_n=y使得f（x_{i+1}，x_i）位于a之下或b之下，对于所有i < n。(iii)子集{f（x，0）|x \in P}生成半格S。此外，P的任何有限有界子集都有一个连接，并且在S有界的情况下P有界。此外，该构造在有限分配（v，0，1）-半格的格指标图上是函子的。
英文摘要：
We prove that for any distributive join-semilattice S, there are a meet-semilattice P with zero and a map f:PxP-->S such that f(x,z)<=f(x,y)vf(y,z) and x<=y implies that f(x,y)=0, for all x,y,z in P, together with the following conditions: (i) f(y,x)=0 implies that x=y, for all x<=y in P. (ii) For all x\leq y in P and all a,b in S, if f(y,x)=avb, then there are a positive integer n and a decomposition x=x_0<=x_1<=...<=x_n=y such that f(x_{i+1},x_i) lies either below a or below b, for all i < n. (iii) The subset {f(x,0)|x\in P} generates the semilattice S. Furthermore, any finite, bounded subset of P has a join, and P is bounded in case S is bounded. Furthermore, the construction is functorial on lattice-indexed diagrams of finite distributive (v,0,1)-semilattices.
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