A duality for finite lattices
Résumé
A presentation is a triple $\leangle X,\leq,M\rangle$ with $\langle X,\leq\rangle$ a finite poset and $M : X \rTo \P(\P(X))$ -- these data being subject to additional constraints. Given a presentation we can define closed subsets of $X$, whence a finite lattice. Given a finite lattice $L$, we can define its presentation: $X$ is the set of join-irreducible elements of $L$, $\leq$ is the restriction of the order to join-irreducible elements, and $M(x)$ is the set of minimal join-covers of $x$. Morphisms of presentations can be defined as some zig-zag relations. Our main result is: \emph{the category of presentations is dually equivalent to the category of finite lattice}. The two constructions described above are the object part of contravariant functors giving rise to the duality. We think of presentations as semantic domains for lattice terms and formulas of substructural logics. Relying on previous work by Nation and Semenova, we show that some equational properties of finite lattices correspond to first order properties of presentations. Namely, for each finite tree $T$, we construct an equation $e_{T}$ that holds in a finite lattice if and only its presentation does not have some a shape from $T$. We illustrate further use of these semantics within the theory of fixed points over finite lattices: we generalize, in a non-trivial way, the well known fact that least fixed points on distributive lattice terms can be eliminated.
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