Xin, X., Khan, M., Jun, Y. (2019). Generalized states on EQ-algebras. Iranian Journal of Fuzzy Systems, 16(1), 159-172. doi: 10.22111/ijfs.2019.4491

Xiao Long Xin; M. Khan; Y. B. Jun. "Generalized states on EQ-algebras". Iranian Journal of Fuzzy Systems, 16, 1, 2019, 159-172. doi: 10.22111/ijfs.2019.4491

Xin, X., Khan, M., Jun, Y. (2019). 'Generalized states on EQ-algebras', Iranian Journal of Fuzzy Systems, 16(1), pp. 159-172. doi: 10.22111/ijfs.2019.4491

Xin, X., Khan, M., Jun, Y. Generalized states on EQ-algebras. Iranian Journal of Fuzzy Systems, 2019; 16(1): 159-172. doi: 10.22111/ijfs.2019.4491

^{2}Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan

^{3}Department of Mathematics Education, Gyeongsang National University, Jinju 660-701, Korea

Abstract

In this paper, we introduce a notion of generalized states from an EQ-algebra E1 to another EQ-algebra E2, which is a generalization of internal states (or state operators) on an EQ-algebra E. Also we give a type of special generalized state from an EQ-algebra E1 to E1, called generalized internal states (or GI-state). Then we give some examples and basic properties of generalized (internal) states on EQ-algebras. Moreover we discuss the relations between generalized states on EQ-algebras and internal states on other algebras, respectively. We obtain the following results: (1) Every state-morphism on a good EQ-algebra E is a G-state from E to the EQ-algebra E0 = ([0,1],∧0,⊙0,∼0,1). (2) Every state operator µ satisfying µ(x)⊙µ(y) ∈ µ(E) on a good EQ-algebra E is a GI-state on E. (3) Every state operator τ on a residuated lattice (L,∧,∨,⊙,→,0,1) can be seen a GI-state on the EQ-algebra (L,∧,⊙,∼,1), where x ∼ y := (x → y) ∧ (y → x). (4) Every GI-state σ on a good EQ-algebra (L,∧,⊙,∼,1) is a internal state on equality algebra (L,∧,∼,1). (5) Every GI-state σ on a good EQ-algebra (L,∧,⊙,∼,1) is a left state operator on BCK-algebra (L,∧,→,1), where x → y = x ∼ x∧y.

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