Omega Algebras and Regular Equations

We study a weak variant of omega algebra, where one of the usual star induction axioms is absent, in the context of recursive regular equations. We present abstract conditions for explicitly defining the omega operation and use them for proving an algebraic variant of Arden’s rule for solving such equations. We instantiate these results in concrete models—languages, traces and relations—showing, for instance, that the omega captures precisely the empty word property in regular languages. Finally, we derive Salomaa’s axioms for the algebra of regular events. This yields a sound and complete axiomatisation in which the “regular” axioms are weaker than Kleene algebra.