Relational Representation Theorem for Powerset Quantales

(Q,≦,∨) is a complete join semilattice (Q,・,e) is a monoid (∨S)・a=∨{b・a | b∈S} for a∈Q, S⊆Q a・(∨S)=∨{a・b | b∈S} for a∈Q, S⊆Q 2 Background: An Example of Quantale (Rel(A),≦,∨,・,e) is a quantale where Rel(A) is the set of all binary relations on A ≦ is the inclusion ∨ is the union ・ is the composition Q・R={ (a,c) | ∃b. (a,b)∈Q, (b,c)∈R } e is the identity relation 3 Our Goal: Relational Representation Theorem Goal is to give a sufficient condition for a quantale Q to be isomorphic to a subquantale of Rel(A) for some set A. In other words, to give a sufficient condition for a quantale Q to have an injective homomorphism of quantales from Q to Rel(A) for some set A.