Modal correspondence theory on quantales
Alessandra Palmigiano

In [1], the Kripke semantics of basic modal logic has been generalized 
via an embedding of Kripke structures into the larger class of pointed 
stably supported quantales. Independently of this embedding,  completeness 
w.r.t. this quantale-based semantics has been proved for a number of 
logical systems which include K, T, K4, S4 and S5.

Building on the observation that a Sahlqvist correspondence argument 
underlies each of these completeness results, we develop a three-sided 
Sahlqvist-style correspondence theory, involving a positive modal logic 
language, its associated first-order correspondence language and fragments 
of the language of stably supported quantales; we will also discuss open 
problems and further directions of research.

[1] S. Marcelino, P. Resende, An algebraic generalization of Kripke structures, 
Mathematical Proceedings of the Cambridge Philosophical Society (2008), 145 : 549-577.