Dual Adjunction Between $\Omega$-Automata and Wilke Algebra Quotients

TL;DR

This paper establishes a dual adjunction between $\Omega$-automata and Wilke algebra quotients, introducing lasso semigroups as a generalization to better understand $\omega$-regular languages and their automata representations.

Contribution

It introduces lasso semigroups as a new algebraic structure and proves a dual adjunction linking $\Omega$-automata with Wilke algebra quotients, advancing the algebraic understanding of $\omega$-regular languages.

Findings

Lasso semigroups generalize Wilke algebras.

Finite lasso semigroups characterize regular lasso languages.

A dual adjunction is established between automata and algebraic quotients.

Abstract

$\Omega$-automata and Wilke algebras are formalisms for characterising $\omega$-regular languages via their ultimately periodic words. $\Omega$-automata read finite representations of ultimately periodic words, called lassos, and they are a subclass of lasso automata. We introduce lasso semigroups as a generalisation of Wilke algebras that mirrors how lasso automata generalise $\Omega$-automata, and we show that finite lasso semigroups characterise regular lasso languages. We then show a dual adjunction between lasso automata and quotients of the free lasso semigroup with a recognising set, and as our main result we show that this dual adjunction restricts to one between $\Omega$-automata and quotients of the free Wilke algebra with a recognising set.