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Article Dans Une Revue International Journal of Algebra and Computation Année : 2008

Poset representations of distributive semilattices

Résumé

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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Dates et versions

hal-00016421 , version 1 (03-01-2006)
hal-00016421 , version 2 (13-02-2006)
hal-00016421 , version 3 (23-11-2007)

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Friedrich Wehrung. Poset representations of distributive semilattices. International Journal of Algebra and Computation, 2008, 18 (2), pp.321--356. ⟨10.1142/S0218196708004469⟩. ⟨hal-00016421v3⟩
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