# Kleisli, Parikh and Peleg Compositions and Liftings for Multirelations

## TL;DR

This paper formalizes various compositions and liftings of multirelations, revealing their algebraic properties and conditions under which they form categories, advancing the semantic modeling of nondeterministic systems.

## Contribution

It provides relational formalizations of Kleisli, Parikh, and Peleg compositions for multirelations, analyzing their algebraic properties and categorical structures.

## Key findings

- Kleisli composition is associative but may lack units.
- Parikh composition is non-associative and unitless, forming a category on up-closed multirelations.
- Peleg composition has units but is not necessarily associative, forming a category on union-closed multirelations.

## Abstract

Multirelations provide a semantic domain for computing systems that involve two dual kinds of nondeterminism. This paper presents relational formalisations of Kleisli, Parikh and Peleg compositions and liftings of multirelations. These liftings are similar to those that arise in the Kleisli category of the powerset monad. We show that Kleisli composition of multirelations is associative, but need not have units. Parikh composition may neither be associative nor have units, but yields a category on the subclass of up-closed multirelations. Finally, Peleg composition has units, but need not be associative; a category is obtained when multirelations are union-closed.