Poset representations of distributive semilattices

Details Poset representations of distributive semilattices Poset representations of distributive semilattices

2006-01-04

arxiv.org/abs/math/0601058v3

International Journal of Algebra and Computation 18, 2 (2008) 321--356

Mathematics

Rings and Algebras

To appear in Internat. J. Algebra Comput

Scientific paper

10.1142/S0218196708004469

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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