  $\Omega$-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.
