A suplattice is a poset which has all joins (and in particular is a join-semilattice). By the adjoint functor theorem for posets, a suplattice necessarily has all meets as well and so is a complete lattice. However, a suplattice homomorphism preserves joins, but not necessarily meets. Furthermore, a large semilattice which has all small joins need not have all meets, but might still be considered a large suplattice (even though it may not even be a lattice).

Dually, an inflattice is a poset which has all meets, and an inflattice homomorphism in a monotone function that preserves all meets.

A frame (dual to a locale) is a suplattice in which finitary meets distribute over arbitrary joins. (Frame homomorphisms preserve all joins and finitary meets.)

The categorySupLat of suplattices and suplattice homomorphisms admits a tensor product which represents “bilinear maps,” i.e. functions which preserve joins separately in each variable. Under this tensor product, the category of suplattices is a star-autonomous category in which the dualizing object is the suplattice dual to the object $TV$ of truth-values. A semigroup in this monoidal category is a quantale, including frames as a special case when the quantale is idempotent and unital. Modules over them are modules over quantales (quantic modules with special case of localic modules, used in the localic analogue of the Grothendieck’s descent theory in Joyal and Tierney).

André Joyal, M. Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 51 (1984), no. 309, vii+71 pp.

Last revised on April 10, 2018 at 12:29:15.
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