Finite many-valuedness is more than just an extension or generalization of two-valuedness.

Many-valuedness involves relations between values as well as operations on values. Binary relations suffice and mostly requires transitivity, and binary operations suffice as mostly requiring associativity. Partial orders giving (finite) lattices are then useful since order as relation reshapes into lattice operations.

A binary operation in a semigroup, distributing over an order structure as its counterpart is the starting point for quantales as Semigroups in Complete Lattices.

Even for smaller numbers of elements in base sets, the number of quantales is very large, where the Catalogue of Quantales supports the view on Practicability of Quantales.

Contact: Patrik Eklund