AUTONOMOUS POSETS AND QUANTALES

In this paper we consider partially ordered algebraic structures arisi ng in the semantics of formulas of a non commutative version of Girard linear logic. The non commutative version we treat is the one recentl y proposed by V. M. Abrusci. We introduce autonomous quantales and pro ve a completion theorem from autonomous posets to autonomous quantales and a representation theorem ''every autonomous quantale is isomorphi c to a non commutative phase space quantale'', generalizing previous e xisting results valid in the commutative case.