Each Girard quantale (i.e., commutative quantale with a selected dualizing element) provides a support for a semantics for linear propositional formulas (but not for linear derivations). Several constructions of Girard quantales are known. We give two more constructions, one using an arbitrary partially ordered monoid and one using a partially ordered group (both commutative). In both cases the semantics can be controlled be a relation between pairs of elements of the support and formulas. This gives us a neat way of handling duality.
"Monoid based semantics for linear formulas." J. Symbolic Logic 67 (2) 505 - 527, June 2002. https://doi.org/10.2178/jsl/1190150094