Convolution Algebras: Relational Convolution, Generalised Modalities and Incidence Algebras

Abstract

Convolution is a ubiquitous operation in mathematics and computing. TheKripke semantics for substructural and interval logics motivates its study forquantale-valued functions relative to ternary relations. The resulting notionof relational convolution leads to generalised binary and unary modal operatorsfor qualitative and quantitative models, and to more conventional variants,when ternary relations arise from identities over partial semigroups.Convolution-based semantics for fragments of categorial, linear and incidence(segment or interval) logics are provided as qualitative applications.Quantitative examples include algebras of durations and mean values in theduration calculus