Quantale

Summary

A quantale is a complete lattice with an associative multiplication that distributes over arbitrary joins on both sides. Lattices of ideals of a ring, frames of open sets, and power sets of monoids are quantales. Quantales generalise frames and locales to a non-commutative or non-idempotent setting. They appear in noncommutative topology and linear logic. The multiplication of a quantale always has left and right residuals, which links quantales with residuated lattices.