Dual adjunction between Ω-automata and Wilke algebra quotients

Ω-automata and Wilke algebras are formalisms for characterising ω-regular languages via their ultimately periodic words. Ω-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 Ω-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 Ω-automata and quotients of the free Wilke algebra with a recognising set.