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Autonomous posets and quantales. (English) Zbl 0803.06016

Summary: We consider partially ordered algebraic structures arising in the semantics of formulas of a non-commutative version of Girard linear logic. The non-commutative version we treat is the one recently proposed by V. M. Abrusci. We introduce autonomous quantales and prove a completion theorem from autonomous posets to autonomous quantales and a representation theorem “every autonomous quantale is isomorphic to a non-commutative phase space quantale”, generalizing previous existing results valid in the commutative case.

MSC:

06F05 Ordered semigroups and monoids
03G25 Other algebras related to logic
03B20 Subsystems of classical logic (including intuitionistic logic)
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