# Convolution Algebras: Relational Convolution, Generalised Modalities and Incidence Algebras

## TL;DR

This paper explores convolution operations in mathematical logic and computing, introducing relational convolution for quantale-valued functions, leading to new modal operators and applications in various logical and quantitative models.

## Contribution

It develops a unified framework for relational convolution in logic, generalizing modal operators and applying to categorial, linear, and interval logics with quantitative examples.

## Key findings

- Relational convolution leads to new modal operators for qualitative and quantitative models.
- Convolution-based semantics are provided for fragments of various logics.
- Applications include algebras of durations and mean values in duration calculus.

## Abstract

Convolution is a ubiquitous operation in mathematics and computing. The Kripke semantics for substructural and interval logics motivates its study for quantale-valued functions relative to ternary relations. The resulting notion of relational convolution leads to generalised binary and unary modal operators for 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 the duration calculus.