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Monitors are a key tool in the field of runtime verification, where they are used to verify system properties by analyzing execution traces generated by processes. Work on runtime monitoring carried out in a series of papers by Aceto et al.$~$has specified monitors using a variation on the regular fragment of Milner's CCS and studied two trace-based notions of equivalence over monitors, namely verdict and $ω$-verdict equivalence. This article is devoted to the study of the equational logic of monitors modulo those two notions of equivalence. It presents complete equational axiomatizations of verdict and $ω$-verdict equivalence for closed and open terms over recursion-free monitors. It is also shown that verdict equivalence has no finite equational axiomatization over open monitors when the set of actions is finite and contains at least two actions.