Catoids and Modal Convolution Algebras

Authors

Uli Fahrenberg, Christian Johansen, Georg Struth, Krzysztof Ziemiański

Journal

Algebra Universalis

Type

article

Projects

AA"AA" is not in the list (Vaucanson, Spot, URBI, Olena, APMC, Tiger, Climb, Speaker ID, Transformers, Bison, ...) of allowed values for the "Related project" property.

Date

2023-03-05

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,
  url		= {https://link.springer.com/article/10.1007/s00012-023-00805-9},
  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.}
}