We show that every state-morphism operator on an equality algebra is an internal state operator on it and prove that the converse is correct for the linearly ordered equality algebras under a special condition. Then we show that there is a one-to-one correspondening between congruence relations on a state-morphism (linearly ordered state) equality algebra and statemorphism (state) deductive systems on it. Moreover, we define the notion of homomorphism on equality algebras and we investigate the relation between state operators and state-morphism operators with equality-homomorphism. Finally, we characterize the simple and semisimple classes of state-morphism equality algebras.