A state operator on the uE-algebra

In this paper, we introduce the variety of equality algebras endowed with universal quantifiers (called uE-algebras), and establish an equivalent category for the corresponding category. Most significantly, we present an expansion of uE-algebras by enriching them with an internal state operator (which are referred to as state uE-algebras), and investigate some of their fundamental algebraic properties. In particular, we discuss the correspondence between state m-deductive systems and m-congruences of state uE-algebras. We also provide a set of characterizations for subdirectly irreducible state uE-algebras, together with characterizations for local and simple state-morphism uE-algebras, which play a crucial role in studying the variety of state uE-algebras. Finally, using the notion of adjoint m-pairs, we obtain a representation for state-morphism operators on uE-algebras.