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Abstract

In this paper, we introduce the notion of a generalized state operator by extending the codomain of a state operator to a more general algebraic structure, that is, from a BCI-algebra X to an arbitrary BCI-algebra Y. Also we give some special types of generalized state operators according to the structures of Y such as GD-states, GK-states and state operators. We get that generalized state operators are the generalization of state-morphisms and state-morphism operators on BCK/BCI-algebras. Then we focus on studying state operators on BCI-algebras and obtain some important results: (1) State BCI-algebras are a genuine generation of state MV-algebras. (2) There is a bijection between all state congruences and all closed state ideals on a state BCI-algebra. (3) If (X, σ) is a nontrivial subdirectly irreducible state BCI-algebra with Ker (σ) =0, then σ (X) is a nontrivial subdirectly irreducible subalgebra of X. Moreover we give some characterizations of several special types of BCI-algebras by use of state operators and differential states.

Keywords

BCI-algebra generalized state operator state operator state ideal differential state

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Generalized state operators on BCI-algebras

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