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In this paper, we present some characterizations of prime and maximal filters. Moreover, we introduce ∩-irreducible filters of an equality algebra, investigate some results about them and relations between maximal, prime, V-irreducible and ∩-irreducible filters in equality algebra. Also, we introduce spec- trum of an equality algebra and prove that the spectrum endowed with Zariski topology is a compact T0 topological space and maximal spectrum (as a subspace of that) is a compact T1 topological space. [ABSTRACT FROM AUTHOR]