Equivalence of multiset-based consequence relations

The pioneering work of Blok and Jónsson, and its further development by Galatos and Tsinakis, initiated an abstract study of consequence relations through the lens of module theory, treating consequence relations over all types of syntactic objects on an equal footing. Despite this generality, their framework retains the assumption that premises in a consequence relation form a mere set, rather than a more structured collection. An attempt to extend this framework to account for inferentially substructural generalizations of consequence relations, where the premises have the structure of a finite multiset, was recently made by Cintula, Gil-Férez, Moraschini, and Paoli. In this paper, we propose a different substructural generalization of the Galatos–Tsinakis approach, where the premises are instead taken to form a set of finite multisets. This yields a smoother and more flexible framework that, unlike the approach of Cintula et al., subsumes the original theory of Galatos and Tsinakis as a special case.