# nLab suplattice

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 distrubute over arbitrary joins. (Frame homomorphisms preserve all joins and finitary meets.)

The category SupLat 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 $\mathrm{TV}$ of truth-values. A monoid in this monoidal category is a quantale, including frames as a special case.

Revised on January 31, 2010 17:40:36 by Toby Bartels (173.60.119.197)