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

Anton Chernev; Helle Hvid Hansen; Clemens Kupke

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arXiv · 2407.14115

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

Anton Chernev·Helle Hvid Hansen·Clemens Kupke

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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.

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Anton Chernev, Helle Hvid Hansen, Clemens Kupke. 2024-07-19. Dual Adjunction Between $\Omega$-Automata and Wilke Algebra Quotients. https://arxiv.org/abs/2407.14115

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