Dual Adjunction Between $\Omega$-Automata and Wilke Algebra Quotients
<|reference_start|>Dual Adjunction Between $\Omega$-Automata and Wilke Algebra Quotients: $\Omega$-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.<|reference_end|>
@article{chernev2024dual,
    title={Dual Adjunction Between $\Omega$-Automata and Wilke Algebra Quotients},
    author={Anton Chernev and Helle Hvid Hansen and Clemens Kupke},
    journal={arXiv preprint arXiv:2407.14115},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.14115},
    primaryClass={cs.FL}
}