Recognisable languages over monads

TL;DR

This paper proposes using monads from category theory as a unifying framework to analyze and design algebraic structures for various types of languages, such as words and trees.

Contribution

It introduces monads as a general, abstract tool to unify and extend algebraic language theory across different structures.

Findings

Monads provide a versatile framework for algebraic language theory.

The approach unifies existing theories for words, trees, and other structures.

It facilitates the design of new algebraic systems for languages.

Abstract

The principle behind algebraic language theory for various kinds of structures, such as words or trees, is to use a compositional function from the structures into a finite set. To talk about compositionality, one needs some way of composing structures into bigger structures. It so happens that category theory has an abstract concept for this, namely a monad. The goal of this paper is to propose monads as a unifying framework for discussing existing algebras and designing new algebras.