Generalized state operators on BCI-algebras

Article type: Research Article

Authors: Xin, Xiaolong^{a; *} | Cheng, Xiaoyun^a | Zhang, Xiaohong^b

Affiliations: [a] College of Mathematics, Northwest University, Xi’an, China | [b] Department of Mathematics, Shanghai Maritime University, Shanghai, China

Correspondence: [*] Corresponding author. Xiaolong Xin, School of Mathematics, Northwest University, Xi’an 710127, China. Tel.: +86 13571952880; Fax: +86 2988308430; E-mail: xlxin@nwu.edu.cn.

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

DOI: 10.3233/JIFS-16550

Journal: Journal of Intelligent & Fuzzy Systems, vol. 32, no. 3, pp. 2591-2602, 2017