# Catoids and Modal Convolution Algebras

## Abstract

We show how modal quantales arise as convolution algebras '"UNIQ--postMath-00000001-QINU"' of functions from catoids '"UNIQ--postMath-00000002-QINU"', that ismultisemigroups with a source map '"UNIQ--postMath-00000003-QINU"' and a target map '"UNIQ--postMath-00000004-QINU"', into modal quantales '"UNIQ--postMath-00000005-QINU"', which can be seen as weight or value algebras. In the tradition of boolean algebras with operators we study modal correspondences between algebraic laws in '"UNIQ--postMath-00000006-QINU"', '"UNIQ--postMath-00000007-QINU"' and '"UNIQ--postMath-00000008-QINU"'. The class of catoids we introduce generalises Schweizer and Sklar's function systems and object-free categories to a setting isomorphic to algebras of ternary relations, as they are used for boolean algebras with operators and substructural logics. Our results provide a generic construction of weighted modal quantales from such multisemigroups. It is illustrated by many examples. We also discuss how these results generalise to a setting that supports reasoning with stochastic matrices or probabilistic predicate transformers.

## Bibtex (lrde.bib)

@Article{	  fahrenberg.23.alguniv,
author	= {Uli Fahrenberg and Christian Johansen and Georg Struth and
Krzysztof Ziemia{\'n}ski},
title		= {Catoids and Modal Convolution Algebras},
journal	= {Algebra Universalis},
volume	= 84,
number	= {10},
year		= 2023,
month		= mar,
doi		= {10.1007/s00012-023-00805-9},
abstract	= { We show how modal quantales arise as convolution algebras
$Q^X$ of functions from catoids $X$, that is,
multisemigroups with a source map $\ell$ and a target map
$r$, into modal quantales $Q$, which can be seen as weight
or value algebras. In the tradition of boolean algebras
with operators we study modal correspondences between
algebraic laws in $X$, $Q$ and $Q^X$. The class of catoids
we introduce generalises Schweizer and Sklar's function
systems and object-free categories to a setting isomorphic
to algebras of ternary relations, as they are used for
boolean algebras with operators and substructural logics.
Our results provide a generic construction of weighted
modal quantales from such multisemigroups. It is
illustrated by many examples. We also discuss how these
results generalise to a setting that supports reasoning
with stochastic matrices or probabilistic predicate
transformers.}
}