Cyclic system for an algebraic theory of alternating parity automata

$\omega$-regular languages are a natural extension of the regular languages to the setting of infinite words. Likewise, they are recognised by a host of automata models, one of the most important being Alternating Parity Automata (APAs), a generalisation of B\"uchi automata that symmetrises both the transitions (with universal as well as existential branching) and the acceptance condition (by a parity condition). In this work we develop a cyclic proof system manipulating APAs, represented by an algebraic notation of Right Linear Lattice expressions. This syntax dualises that of previously introduced Right Linear Algebras, which comprised a notation for non-deterministic finite automata (NFAs). This dualisation induces a symmetry in the proof systems we design, with lattice operations behaving dually on each side of the sequent. Our main result is the soundness and completeness of our system for $\omega$-language inclusion, heavily exploiting game theoretic techniques from the theory of $\omega$-regular languages.