EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 3, 2017, 455-472 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global States on pseudo-BCI algebras Xiao Long Xin1,∗, Yi Jun Li1, Yu Long Fu2 1 School of Mathematics, Northwest University, Xi’an, China 2 School of Cyber Engnieering, Xidian University, Xi’an, China Abstract. In this paper, we discuss the structure of pseudo-BCI algebras and get that any pseudo- BCI algebra is a union of it’s branches. We introduce the notion of local bounded pseudo-BCI algebras and study some related properties. Moreover we define two operations ∧1, ∧2 in a local bounded pseudo-BCI algebra A and two local operations ∨1 and ∨2 in V (a) for a ∈M(A). We show that in a ∧1(∧2)-commutative local bounded pseudo-BCI algebra A, (V (a),∧1,∨1)((V (a),∧2,∨2)) forms a lattice for all a ∈ M(A). We define a Bosbach state on a local bounded pseudo-BCI algebra. Then we give two examples of local bounded pseudo-BCI algebras to show that there is local bounded pseudo-BCI algebras having a Bosbach state but there is some one having no Bosbach states. Moreover we discuss some basic properties about Bosbach states. If s is a Bosbach state of a local bounded pseudo-BCI algebra A, we prove that A/ker(s) is equivalent to an MV-algebra. We also introduce the notion of state-morphisms on local bounded pseudo-BCI algebras and discuss the relations between Bosbach states and state-morphisms. Finally we give some characterization of Bosbach states. 2010 Mathematics Subject Classifications: 03G25 Key Words and Phrases: Pseudo-BCI algebra, local bounded, state, state-morphism, MV- algebra 1. Introduction BCK/BCI algebras were introduced originally by Iséki in [17] and [18] with a binary operation ∗ modeling the set-theoretical difference. Another motivation is from classical and non-classical propositional calculi modeling logical implications. Such algebras contain as a special subfamily of a family of MV-algebras where some important fuzzy structures can be studied. For more about BCK algebras, see [22]. Pseudo-BCK algebras were originally introduced by Georgescu and Iorgulescu in [13] as algebras with ”two differences”, a left- and right-difference, instead of one ∗ and with a constant element 0 as the least element. In [12], a special subclass of pseudo-BCK algebras, called Lukasiewicz pseudo-BCK algebras, was introduced and it was shown that ∗Corresponding author. Email addresses: xlxin@nwu.edu.cn (X.L. Xin), liyijun2009@126.com (Y.J. Li), ylfu@xidian.edu.cn (Y.L. Fu) http://www.ejpam.com 455 c© 2017 EJPAM All rights reserved. X.L. Xin, Y.J. Li, Y.L. Fu / Eur. J. Pure Appl. Math, 10 (3) (2017), 455-472 456 it is always a subalgebra of the positive cone of some `-group (not necessarily abelian). The class of Lukasiewicz pseudo-BCKalgebras is a variety whereas the class of pseudo- BCKalgebras is not; it is only a quasivariety because it is not closed under homomorphic images. For a guide through the pseudo-BCK algebras realm, see the monograph [16]. In [8], W. A. Dudek and Y. B. Jun introduced the notion of pseudo-BCI algebras as an extension of BCI-algebras, and investigated some properties. MV-algebras entered into mathematics just 50 years ago due to Chang [3], but the notion of a state for MV-algebras was introduced by Mundici [23] in 1995 as averaging of the truth-value in Lukasiewicz logic. BL-algebras were introduced in the 1990s by Hájek [14] as the equivalent algebraic semantics for its basic fuzzy logic. In [5], authors defined a state-operator and a strong state-operator for a BL-algebra and prove some of their basic properties. L. Z. Liu studied the existence of Bosbach states and Riečan states on finite monoidal t-norm based algebras in [21]. Some examples show that there exist MTL-algebras having no Bosbach states and Riečan states. In [10], Dvurečenskij introduced measures and states on BCK-algebras, and showed that the set of elements of measure 0 is an ideal, and the corresponding quotient BCK- algebra is commutative with a lifted original measure. Ciungu and Dvurečenskij [4] ex- tended the notions of measures and states presented in Dvurečen-skij and Pulmannová [9] to the case of pseudo-BCK algebras, studied similar properties, and prove that, under some conditions, the notion of a state in the sense of Dvurečenskij and Pulmannová [9] coincides with the Bosbach state. The aim of this paper is to introduce and study the state theory on local bounded pseudo-BCI algebras. This paper is organized as follows: in Section 2, we recall notions of BCI-algebras and the notion and some properties of pseudo-BCI algebras. In the same time, we discuss the structure of pseudo-BCI algebras and get that any pseudo-BCI algebra is a union of it’s branches. In Section 3, we introduce the notion of local bounded pseudo- BCI algebras and study some related properties. In Section 4, we define a Bosbach state on a local bounded pseudo-BCI algebra. Then we give two examples of local bounded pseudo-BCI algebras to show that there is local bounded pseudo-BCI algebras having a Bosbach state but there is some one having no Bosbach states. Moreover we discuss some of their basic properties. We discuss the relation between local bounded pseudo- BCI algebras and MV -algebras. We also introduce the notion of state-morphisms on local bounded pseudo-BCI algebras and discuss the relations between Bosbach states and state-morphisms. Finally we give some characterization on Bosbach states. 2. Pseudo-BCI algebras Recall that a BCI-algebra is an algebra (X, ∗, 0) of type (2,0) satisfying the following axioms: for every x, y, z ∈ X, (1) ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0, (2) (x ∗ (x ∗ y)) ∗ y = 0, (3) x ∗ x = 0, (4) x ∗ y = 0 and y ∗ x = 0 imply x = y. For any BCI-algebra X, the relation ≤ defined by x ≤ y if and only if x ∗ y = 0 is a partial order on X. A nonempty subset I of a BCI-algebra X is called a BCI- ideal of X if it satisfies (1) 0 ∈ I, (2) For all x, y ∈ X,x ∗ y ∈ I, y ∈ I ⇒ x ∈ I. X.L. Xin, Y.J. Li, Y.L. Fu / Eur. J. Pure Appl. Math, 10 (3) (2017), 455-472 457 We recall the notion and some properties of pseudo-BCI algebras. Definition 1. [19] A pseudo-BCI algebras is a structure A = (A,≤, ∗, ◦, 0), where ≤ is a binary relation on A, ∗ and ◦ are binary operations on A and ”0” is an element of A, satisfying, for all x, y, z ∈ A, (I1) (x ∗ y) ◦ (x ∗ z) ≤ z ∗ y, (x ◦ y) ∗ (x ◦ z) ≤ z ◦ y. (I2) x ∗ (x ◦ y) ≤ y, x ◦ (x ∗ y) ≤ y. (I3) x ≤ x. (I4) x ≤ y and y ≤ x imply x = y. (I5) x ≤ y iff x ∗ y = 0 iff x ◦ y = 0. Definition 2. [13] A pseudo-BCK algebra is a structure A = (A,�,→, , 1) where � is a binary relation on A, and → and are binary operations on A and 1 is an element of A satisfying, for all x, y, z ∈ A, the axioms: (K1) x→ y � (y → z) (x→ z), x y � (y z)→ (x z). (K2) x � (x→ y) y, x � (x y)→ y. (K3) x � x. (K4) x � 1. (K5) if x � y and y � x, then x = y. (K6) x � y iff x→ y = 1 iff x y = 1. Remark 1. (1) A pseudo-BCK algebra A = (A,�,→ , 1) can be seen a pseudo-BCI algebra A = (A,≤, ∗, ◦, 0) if x→ y = y ∗ x, x y = y ◦ x, 1 = 0 and x � y iff y ≤ x for all x, y ∈ A. (2) A pseudo-BCI algebra is a BCI algebra if ∗ = ◦. (3) The relation ≤ is a partial order on a pseudo-BCI algebra A. Now we give two pseudo-BCI algebras which are not pseudo-BCK algebras. Example 1. Let A = {0, u, v, w, t, 1, a, b}. The order of the elements in A is as the following Hasse diagram: c c c c c c c c � �� @ @@ � �� @ @@ 0 u wv t 1 a b Now the operations ∗ and ◦ are defined by Tables 2.1 and 2.2, respectively. Simple calcu- lations show that (A,6, ∗, ◦, 0) is a pseudo-BCI algebra. Example 2. Let A = {0, x, y, z, 1, a, b} in which the order of elements in A is as the following Hasse diagram: X.L. Xin, Y.J. Li, Y.L. Fu / Eur. J. Pure Appl. Math, 10 (3) (2017), 455-472 458 ∗ 0 u v w t 1 a b 0 0 0 0 0 0 0 a a u u 0 0 0 0 0 a a v v v 0 v 0 0 a a w w w w 0 0 0 a a t t t w t 0 0 a a 1 1 1 1 1 1 0 a a a a a a a a a 0 0 b b b b b b a 1 0 Tables 2.1 ◦ 0 u v w t 1 a b 0 0 0 0 0 0 0 a a u u 0 0 0 0 0 a a v v v 0 v 0 0 a a w w w w 0 0 0 a a t t t t v 0 0 a a 1 1 1 1 1 1 0 a a a a a a a a a 0 0 b b b b b b a 1 0 Tables 2.2 c c c c c c c @ @ @ � � � @ @ @ � � � 0 x y z 1 a b Let the operations ∗, ◦ be given by the following Tables 2.3 and 2.4. ∗ 0 x y z 1 a b 0 0 0 0 0 0 a a x x 0 0 0 0 a a y y y 0 y 0 a a z z z z 0 0 a a 1 1 1 1 y 0 a a a a a a a a 0 0 b b a a a a y 0 Tables 2.3 ◦ 0 x y z 1 a b 0 0 0 0 0 0 a a x x 0 0 0 0 a a y y y 0 y 0 a a z z z z 0 0 a a 1 1 1 z 1 0 a a a a a a a a 0 0 b b b a b a x 0 Tables 2.4 Then (A,6, ∗, ◦, 0) is a pseudo-BCI algebra. Proposition 1. [19] In a pseudo-BCI algebras A the following hold: (p1) x ≤ 0⇒ x = 0. (p2) x ≤ y ⇒ z ∗ y ≤ z ∗ x and z ◦ y ≤ z ◦ x. (p3) x ≤ y, y ≤ z ⇒ x ≤ z. (p4) (x ∗ y) ◦ z = (x ◦ z) ∗ y. (p5) x ∗ y ≤ z ⇔ x ◦ z ≤ y. (p6) (x ∗ y) ∗ (z ∗ y) ≤ x ∗ z, (x ◦ y) ◦ (z ◦ y) ≤ x ◦ z. (p7) x ≤ y ⇒ x ∗ z ≤ y ∗ z, x ◦ z ≤ y ◦ z. X.L. Xin, Y.J. Li, Y.L. Fu / Eur. J. Pure Appl. Math, 10 (3) (2017), 455-472 459 (p8) x ∗ 0 = x = x ◦ 0. (p9) x ∗ (x ◦ (x ∗ y)) = x ∗ y, x ◦ (x ∗ (x ◦ y)) = x ◦ y. Proposition 2. [19] In a pseudo-BCI algebra A the following holds for all x, y, z ∈ A: (i) 0 ∗ (x ◦ y) ≤ y ◦ x. (ii) 0 ◦ (x ∗ y) ≤ y ∗ x. (iii) 0 ∗ (x ∗ y) = (0 ◦ x) ◦ (0 ∗ y). (iv) 0 ◦ (x ◦ y) = (0 ∗ x) ∗ (0 ◦ y). Definition 3. [19] An element a of a pseudo-BCI algebra A is called a pseudo-atom if for every x ∈ A, x ≤ a implies x = a. The set of all pseudo-atoms of a pseudo-BCI algebra A is denoted by M(A). Obviously, 0 ∈M(A). Proposition 3. Let A be a pseudo-BCI algebra and a ∈ A. The following conditions are equivalent: (1) a is a pseudo-atom of A; (2) y ∗ (y ◦ a) = a (or y ◦ (y ∗ a) = a) for all y ∈ A; (3) y ∗ (y ◦ (a ∗ x)) = a ∗ x (or y ◦ (y ∗ (a ◦ x)) = a ◦ x) for all x, y ∈ A. Proof. (1) ⇒ (2). By I2, y ∗ (y ◦ a) ≤ a. Since a is a pseudo-atom of A, we have y ∗ (y ◦ a) = a. (2)⇒ (3) Obviously. (3)⇒ (1) It follows from Proposition 3.6 of [19]. By Proposition 3, we have x ∗ (x ◦ a) = x ◦ (x ∗ a) = a for all a ∈M(A) and x ∈ A. Corollary 1. Let A be a pseudo-BCI algebra. Then for all a ∈M(A) and x ∈ A, we have a ∗ x ∈M(A) and a ◦ x ∈M(A). Proof. Let a ∈M(A) and x ∈ A. By Proposition 3.8(3), we have y ∗ (y ◦ (a∗x)) = a∗x for all y ∈ A. Using Proposition 3.8(2), we get that a ∗ x is a pseudo-atom of A, that is a ∗ x ∈M(A). Similarly we can prove a ◦ x ∈M(A). Let A be a pseudo-BCI algebra. For a ∈ M(A), define V (a) = {x ∈ A | a ≤ x}. V(a) is called a branch of A. Obviously a ∈ V (a). Proposition 4. Let A be a pseudo-BCI algebra, a, b ∈ M(A) and a 6= b. Then V (a) ∩ V (b) = ∅. Proof. Assume V (a) ∩ V (b) 6= ∅, then there is x ∈ V (a) ∩ V (b). Hence a ≤ x and b ≤ x. It follows that (b ∗ (b ◦ a)) ◦ (b ∗ (b ◦ x)) ≤ (b ◦ x) ∗ (b ◦ a) ≤ a ◦ x = 0. So (b ∗ (b ◦ a)) ◦ (b ∗ (b ◦ x)) = 0. Hence b ∗ (b ◦ a) ≤ (b ∗ (b ◦ x)) = b. Since b ∈M(A), we have b ∗ (b ◦ a) = b. Note that b = (b ∗ (b ◦ a)) ≤ a. Similarly a ≤ b. By Definition 3.1, we have a = b. It is a contradiction, hence V (a) ∩ V (b) = ∅. X.L. Xin, Y.J. Li, Y.L. Fu / Eur. J. Pure Appl. Math, 10 (3) (2017), 455-472 460 Proposition 5. Let A be a pseudo-BCI algebra and x, y ∈ A. If x ≤ y, then x, y are in the same branch of A. Proof. Assume that x ∈ V (a) and y ∈ V (b) for some a, b ∈ M(A) and a 6= b. Then a ≤ x ≤ y. Hence y ∈ V (a) and so y ∈ V (a) ∩ V (b), a contradiction with Proposition 4. Proposition 6. Let A be a pseudo-BCI algebra and x ∈ V (a) for some a ∈M(A). Then 0 ∗ (0 ◦ x) = a and 0 ◦ (0 ∗ x) = a. Proof. Since 0 ∗ (0 ◦ x) ≤ x, we have 0 ∗ (0 ◦ x) ∈ V (a) by Proposition 5. Hence a ≤ 0 ∗ (0 ◦ x). On the other hand, we have (0 ∗ (0 ◦ x)) ◦ a = (0 ◦ a) ∗ (0 ◦ x) = ((a ∗ x) ◦ a) ∗ (0 ◦ x) = ((a ◦ a) ∗ x) ∗ (0 ◦ x) = (0 ∗ x) ∗ (0 ∗ x) = 0. Therefore 0 ∗ (0 ◦ x) ≤ a. This shows that 0 ∗ (0 ◦ x) = a. Similarly we can prove 0 ◦ (0 ∗ x) = a. Proposition 7. Let A be a pseudo-BCI algebra. Then for any x ∈ A, 0 ∗ (0 ◦ x) ∈M(A) and 0 ◦ (0 ∗ x) ∈M(A). Proof. Let x ∈ A. In order to prove 0 ◦ (0 ∗ x) ∈ M(A), we assume y ≤ 0 ◦ (0 ∗ x). Then y ◦ (0 ◦ (0 ∗ x)) = 0. By (p4) and (p9) of Proposition 3.3, we have (0 ◦ (0 ∗ x)) ∗ y = (0 ∗ y) ◦ (0 ∗ x) = ((y ◦ (0 ◦ (0 ∗ x))) ∗ y) ◦ (0 ∗ x) = ((y ∗ y) ◦ ((0 ◦ (0 ∗ x)))) ◦ (0 ∗ x) = (0 ◦ ((0 ◦ (0 ∗ x)))) ◦ (0 ∗ x). By Proposition 2(iv), 0 ◦ ((0 ◦ (0 ∗ x))) = (0 ∗ 0) ∗ (0 ◦ (0 ∗ x)) = 0 ∗ (0 ◦ (0 ∗ x)) = 0 ∗ x. Hence (0 ◦ (0 ∗ x)) ∗ y = (0 ◦ ((0 ◦ (0 ∗ x)))) ◦ (0 ∗ x) = (0 ∗ x) ◦ (0 ∗ x) = 0. This shows that 0 ◦ (0 ∗ x) ≤ y and hence y = 0 ◦ (0 ∗ x). Similarly we can prove 0 ∗ (0 ◦ x) ∈M(A). Corollary 2. Let A be a pseudo-BCI algebra. Then for any x ∈ A, (0 ◦ x) ∈ M(A) and (0 ∗ x) ∈M(A). Proof. Since 0 ∗ x = 0 ∗ (0 ◦ (0 ∗ x)) and 0 ◦ x = 0 ◦ (0 ∗ (0 ◦ x)), we have 0 ∗ x ∈M(A) and 0 ◦ x ∈M(A) by Proposition 7. By Propositions 6 and 7, we have 0 ∗ (0 ◦x) = 0 ◦ (0 ∗x) ∈M(A) for all x ∈ A. Denote ax = 0 ∗ (0 ◦ x) = 0 ◦ (0 ∗ x), for x ∈ A. Then ax ∈M(A) and x ∈ V (ax). Using above arguments we can get the structure of a pseudo-BCI algebra. Theorem 1. Let A be a pseudo-BCI algebra. Then {V (a) | a ∈M(A)} forms a partition of A, that is, A = ∪a∈M(A)V (a) and V (a) ∩ V (b) = ∅ for all a, b ∈M(A) and a 6= b. 3. Local bounded pseudo-BCI algebras Let A be a pseudo-BCI algebra. For a ∈M(A), if there is an element 1a ∈ V (a) \ {a} such that for all x ∈ V (a), x ≤ 1a, then 1a is called the local unit of V (a). Note that 1a is unique. X.L. Xin, Y.J. Li, Y.L. Fu / Eur. J. Pure Appl. Math, 10 (3) (2017), 455-472 461 Definition 4. Let A be a pseudo-BCI algebra. If for every a ∈ M(A), V (a) has a local unit, then A is called a local bounded pseudo-BCI algebra. For convenience we denote it by lbp-BCI algebra. Note that the pseudo-BCI algebras given in Examples 1 and 2 are local bounded pseudo-BCI algebras. In Examples 1, M(A) = {0, a}, V (0) = {0, u, v, w, t, 1}, 10 = 0, V (a) = {a, b}, 1a = b. In Examples 2, M(A) = {0, a}, V (0) = {0, x, y, z, 1}, 10 = 0, V (a) = {a, b}, 1a = b. In the following, A shall mean a lbp-BCI algebra unless otherwise specified. We define two negations, − and ∼, as follows: for a ∈M(A) and x ∈ V (a), x− . = 1a ∗ x, x∼ . = 1a ◦ x. Proposition 8. For all x, y ∈ A, we have (1) x−∼ ≤ x, x∼− ≤ x. (2) x ≤ y ⇒ y− ≤ x−, y∼ ≤ x∼. (3) x− = x−∼−, x∼ = x∼−∼. Proof. (1) By (I2) of Definition 1, we have x−∼ ≤ x and x∼− ≤ x. (2) Let x ≤ y, then x, y ∈ V (a) for some a ∈M(A). Hence (1a◦y)∗(1a◦x) ≤ x◦y = 0, and so (1a ◦ y) ∗ (1a ◦ x) = 0. It follows that 1a ◦ y ≤ 1a ◦ x, or y∼ ≤ x∼. Similarly we can prove y− ≤ x−. (3) By (1), we have x∼− ≤ x. Replace x by x−, we get x−∼− ≤ x−. On the other hand, x−∼ ≤ x implies x− ≤ x−∼− by (2). So x− = x−∼−. Similarly we can prove x∼ = x∼−∼. Let A be a pseudo-BCI algebra. For any x, y ∈ A, define x ∧1 y . = y ◦ (y ∗ x), x ∧2 y . = y ∗ (y ◦ x). Proposition 9. In A the following properties hold: (1) ax ∧1 x = x ∧1 ax = ax and ax ∧2 x = x ∧2 ax = ax for all x ∈ A. (2) x ≤ y implies y ∧1 x = x and y ∧2 x = x. (3) x ∧1 x = x and x ∧2 x = x. (4) If x1 ≤ x2, then x1 ∧1 y ≤ x2 ∧1 y and x1 ∧2 y ≤ x2 ∧2 y. Proof. (1) By Proposition 3, we have ax ∧1 x = x ◦ (x ∗ ax) = ax since ax ∈ M(A). Note that for x ∈ V (ax), we get x ∧1 ax = ax ◦ (ax ∗ x) = ax ◦ 0 = ax. So we shows that ax ∧1 x = x ∧1 ax = ax. Similarly we can prove ax ∧2 x = x ∧2 ax = ax for all x ∈ A. (2) Let x ≤ y. Then y∧1 x = x◦ (x∗y) = x◦0 = x and y∧2 x = x∗ (x◦y) = x∗0 = x. (3) We have x ∧1 x = x ◦ (x ∗ x) = x and x ∧2 x = x ∗ (x ◦ x) = x. (4) Let x1 ≤ x2. Note that (x1 ∧1 y) ∗ (x2 ∧1 y) = (y ◦ (y ∗ x1)) ∗ (y ◦ (y ∗ x2)) ≤ (y ∗ x2) ◦ (y ∗ x1) ≤ x1 ∗ x2 = 0. We get x1 ∧1 y ≤ x2 ∧1 y. Similarly we can prove x1 ∧2 y ≤ x2 ∧2 y. Proposition 10. In A the following properties hold for all a ∈M(A) and x, y ∈ V (a): (1) x ∧1 y−∼ = x−∼ ∧1 y−∼ and x ∧2 y∼− = x∼− ∧2 y∼−. (2) x ∧1 y∼ = x−∼ ∧1 y∼ and x ∧2 y− = x∼− ∧2 y−. X.L. Xin, Y.J. Li, Y.L. Fu / Eur. J. Pure Appl. Math, 10 (3) (2017), 455-472 462 Proof. (1) Using Proposition 1, we have y−v∗x = (1a◦(1a∗y))∗x = (1a∗x)◦(1a∗y) = (1a ∗ (1a ◦ (1a ∗ x))) ◦ (1a ∗ y) = (1a ◦ (1a ∗ y)) ∗ (1a ◦ (1a ∗ x)) = y−v ∗ x−∼. Thus x ∧1 y−∼ = y−∼ ◦ (y−∼ ∗ x) = y−∼ ◦ (y−v ∗ x−∼) = x−∼ ∧1 y−v. (2) By Proposition 8 and (1), we get x ∧1 y∼ = x ∧1 (y∼)−∼ = x−∼ ∧1 (y∼)−∼ = x−∼ ∧1 y∼. Proposition 11. In A the following properties hold for all x, y ∈ A: y ∗ (x ∧1 y) = y ∗ x and y ◦ (x ∧2 y) = y ◦ x. Proof. By Proposition 1, we have y ∗ (x∧1 y) = y ∗ (y ◦ (y ∗x) = y ∗x and y ◦ (x∧2 y) = y ◦ (y ∗ (y ◦ x)) = y ◦ x. Proposition 12. Let a ∈M(A). If x, y ∈ V (a), then x ∗ y ∈ V (0) and x ◦ y ∈ V (0). Proof. Using Proposition 2 and 6, we get 0 ◦ (0 ∗ (x ∗ y)) = 0 ◦ ((0 ◦ x) ◦ (0 ∗ y)) = (0∗ (0◦x))∗ (0◦ (0∗y)) = a∗a = 0. Since by (I2) 0◦ (0∗ (x∗y)) ≤ x∗y, we have 0 ≤ x∗y, and so x ∗ y ∈ V (0). Similarly we can prove x ◦ y ∈ V (0). Proposition 13. In A the following properties hold for all a ∈M(A), x, y ∈ V (a): (1) x ∧1 y (y ∧1 x) is a lower bound of {x, y}. (2) x ∧2 y (y ∧2 x) is a lower bound of {x, y}. Proof. By Definition 3.1, we have x∧1 y = y ◦ (y ∗x) ≤ x. Moreover by Proposition 12, y ∗ x ∈ V (0) and so 0 ◦ (y ∗ x) = 0, and (y ◦ (y ∗ x)) ∗ y = (y ∗ y) ◦ (y ∗ x) = 0 ◦ (y ∗ x) = 0. It follows that x ∧1 y = y ◦ (y ∗ x) ≤ y. Similarly we can get that (y ∧1 x) is also a lower bound of {x, y}. (2) Similar to the proof of (1). Definition 5. (1) If for all a ∈M(A) and x, y ∈ V (a), x∧1 y = y ∧1 x, we call A to be a local ∧1-commutative pseudo-BCI algebra. (2) If for all a ∈ M(A) and x, y ∈ V (a), x ∧2 y = y ∧2 x, we call A to be a local ∧2- commutative pseudo-BCI algebra. (3) If A is local ∧1-commutative and local ∧2-commutative, we call A to be local commu- tative. Proposition 14. (1) If A is local ∧1-commutative, then (V (a),∧1) forms a lower simi- lattice for all a ∈M(a). (2) If A is local ∧2-commutative, then (V (a),∧2) forms a lower similattice for all a ∈ M(a). Proof. (1) It needs only to prove that x ∧1 y is the greatest lower bound of {x, y} for all a ∈M(A) and x, y ∈ V (a). Assume that m is a lower bound of {x, y}. We have m∗(x∧1y) = (m◦(m∗y))∗(y◦(y∗x)) = (y∧1m)∗(y◦(y∗x)) = (m∧1y)∗(y◦(y∗x)) = (y ◦ (y ∗m)) ∗ (y ◦ (y ∗ x)) ≤ (y ∗ x) ◦ (y ∗m) ≤ m ∗ x = 0, and so m ≤ (x ∧1 y). X.L. Xin, Y.J. Li, Y.L. Fu / Eur. J. Pure Appl. Math, 10 (3) (2017), 455-472 463 (2) Similar to the proof of (1). For a lbp-BCI algebra A, we can define the following operations in V (a), x ∨1 y = 1a ◦ ((1a ∗ x) ∧1 (1a ∗ y)), x ∨2 y = 1a ∗ ((1a ◦ x) ∧2 (1a ◦ y)), for all a ∈M(A) and for all x, y ∈ V (a). Proposition 15. Let A be a lbp-BCI algebra. (1) If A is local ∧1-commutative, then (V (a),∧1,∨1) forms a lattice for all a ∈M(a). (2) If A is local ∧2-commutative, then (V (a),∧2,∨2) forms a lattice for all a ∈M(a). Proof. (1) Let a ∈ M(A) and x, y ∈ V (a). Since A is local ∧1-commutative, then x = x ◦ (x ∗ 1a) = 1a ◦ (1a ∗ x) ≤ 1a ◦ ((1a ∗ x)∧1 (1a ∗ y)) = x∨1 y. Similarly we can prove y ≤ x ∨1 y. If z ≥ x and z ≥ y, then z ∈ V (a), 1a ∗ x ≥ 1a ∗ z and 1a ∗ y ≥ 1a ∗ z. By Proposition 14, we have 1a ∗ z ≤ (1a ∗ x) ∧1 (1a ∗ y). Therefore x ∨1 y = 1a ◦ ((1a ∗ x) ∧1 (1a ∗ y)) ≤ 1a ◦ (1a ∗ z) = z ◦ (z ∗ 1a) = z. It follows that x ∨1 y is the least upper bound of {x, y}. Applying Proposition 14, we get (V (a),∧1,∨1) forms a lattice. (2) Similar to the proof of (1). Definition 6. Let A be a pseudo-BCI algebra. (1) If for all x, y ∈ A, x∧1 y = y ∧1 x, we call A to be ∧1-commutative. (2) If for all x, y ∈ A, x ∧2 y = y ∧2 x, we call A to be ∧2-commutative. (3) If A is ∧1-commutative and ∧2-commutative, we call A to be sup-commutative. The following result shows that ∧1-commutative (∧2-commutative) pseudo-BCI alge- bras must be pseudo-BCK algebras. Proposition 16. Let A be a pseudo-BCI algebra. Then the following are equivalent: (1) A is ∧1-commutative (∧2-commutative). (2) A is a ∧1-commutative (∧2-commutative) pseudo-BCK algebra. Proof. (1) ⇒ (2). Let A be ∧1-commutative. Then for any a ∈ M(A), we have a∧10 = 0∧1a. Note that a∧10 = 0◦(0∗a) = a by Proposition 6 and 0∧1a = a◦(a∗0) = 0. This shows that a = 0, that is A = V (0). Thus A is a ∧1-commutative pseudo-BCK algebra. Similarly we can prove the result for case of ∧2-commutative. (2)⇒ (1). It is straightforward. Proposition 17. [15] If A is a sup-commutative pseudo-BCK algebra, then ∧1 = ∧2. By Proposition 16 and 17, we can get a characterization of sup-commutative pseudo- BCI algebras. Proposition 18. Let A be a pseudo-BCI algebra. Then the following are equivalent: (1) A is a sup-commutative pseudo-BCI algebra. (2) A is a sup-commutative pseudo-BCK algebra. X.L. Xin, Y.J. Li, Y.L. Fu / Eur. J. Pure Appl. Math, 10 (3) (2017), 455-472 464 4. States on local bounded pseudo-BCI algebras Definition 7. Let A be a lbp-BCI algebra. A Bosbach state on A is a function s : A → [0, 1] such that the following conditions hold: (1) s(x) + s(y ∗ x) = s(y) + s(x ∗ y), for all x, y ∈ A, (2) s(x) + s(y ◦ x) = s(y) + s(x ◦ y), for all x, y ∈ A, (3) s(a) = 1 and s(1a) = 0 where a ∈M(A) and 1a is the local unit of V (a). Example 3. Consider the local bounded pseudo-BCI algebra A given in Example 1. Define the function s : A → [0, 1] by s(0) = 1, s(u) = 1, s(v) = 1, s(w) = 1, s(t) = 1, s(1) = 0, s(a) = 1, s(b) = 0. Then s is a unique Bosbach state on A. Example 4. Consider the local bounded pseudo-BCI algebra A given in Example 2. Define a function s : A→ [0, 1] as follows: s(0) = 1, s(x) = α, s(y) = β, s(z) = γ, s(1) = 0, s(a) = 1, s(b) = 0. Using s(u) + s(v ∗ u) = s(v) + s(u ∗ v), taking u = x, v = 1, u = y, v = 1 and u = z, v = 1, respectively, we get α = 1, β = 1, γ = 0. On the other hand, taking u = z, v = 1 in s(u) + s(v ◦ u) = s(v) + s(u ◦ v), we get γ + 0 = 0 + 1, so 0 = 1 which is a contradiction. Hence A does not admit a Bosbach state. Proposition 19. Let A be a lbp-BCI algebra and s a Bosbach state on A. Then the following properties hold for all x, y ∈ A: (1) If x ≤ y, then s(y ∗ x) = 1 + s(y)− s(x) = s(y ◦ x) and s(y) ≤ s(x). (2) If x, y are in same branch, then s(x ∧1 y) = s(y ∧1 x), s(x ∧2 y) = s(y ∧2 x). (3) If x, y are in same branch, then s(x∧1 y−∼) = s(x−∼ ∧1 y−∼), s(x∧2 y∼−) = s(x∼−∧2 y∼−). (4) If x, y are in same branch, then s(x−∼∧1 y) = s(x∧1 y−∼), s(x∼−∧2 y) = s(x∧2 y∼−). (5) s(x−∼) = s(x) = s(x∼−). (6) s(x−) = 1− s(x) = s(x∼). Proof. (1) Let x ≤ y. It follows from Definition 5.1 that s(y ∗ x) = 1 + s(y)− s(x) = s(y ◦ x). Moreover s(x)− s(y) = 1− s(y ∗ x) ≥ 0 and hence s(y) ≤ s(x). (2) By Proposition 1, we have y ∗ x = y ∗ (x∧1 y). Since x, y are in same branch, then x ∧1 y ≤ x, y by proposition 13. By property (1), we have s(y ∗ x) = s(y ∗ (x ∧1 y)) = 1+s(y)−s(x∧1 y) and s(x∗y) = s(x∗ (y∧1 x)) = 1+s(x)−s(y∧1 x). Using condition (1) from Definition 7 we get s(x∧1 y) = s(y∧1x). Similarly we can prove s(x∧2 y) = s(y∧2x). (3) It follows from Proposition 10. (4) It follows from (2) and (3). (5) For x ∈ A, there is a ∈ M(A) such that x ∈ V (a). Note that x−∼ = x ∧1 1a. By (2), we have s(x−∼) = s(x ∧1 1a) = s(1a ∧1 x) = s(x ◦ (x ∗ 1a)) = s(x). In a similar way, we can prove s(x) = s(x∼−). (6) By (1), we have s(x−) = s(1a ∗ x) = 1 + s(1a)− s(x) = 1− s(x). In a similar way we can get s(x∼) = 1− s(x). X.L. Xin, Y.J. Li, Y.L. Fu / Eur. J. Pure Appl. Math, 10 (3) (2017), 455-472 465 Proposition 20. Let A be a lbp-BCI algebra and s be a Bosbach state on A. Then the following properties hold for all a ∈M(A) and x, y ∈ V (a): (1) s(y ∗ x−∼) = s(y−∼ ∗ x), s(y ◦ x∼−) = s(y∼− ◦ x). (2) s(y−∼ ∗ x) = s(x− ◦ y−) = s(y−∼ ∗ x−∼) = s(y ∗ x−∼), s(y∼− ◦ x) = s(x∼ ∗ y∼) = s(y∼− ◦ x∼−) = s(y ◦ x∼−). (3) s(y−∼ ∗ x∼) = s(y ∗ x∼), s(y∼− ◦ x−) = s(y ◦ x−). Proof. (1) Note that s(y ∗ x−∼) + s(y ◦ (y ∗ x−∼)) = s(y) + s((y ∗ x−∼) ◦ y), or s(y ∗ x−∼) + s(x−∼ ∧1 y) = s(y) + s((y ∗ x−∼) ◦ y). By Proposition 19(4), we have s(y∗x−∼)+s(x∧1y−∼) = s(y)+s((y∗x−∼)◦y) = s(y)+s((y◦y)∗x−∼) = s(y)+s(0∗x−∼). Using Corollary 2, we get 0 ∗ x−∼ ∈ M(A), and so s(0 ∗ x−∼) = 1. Thus s(y ∗ x−∼) = s(y)+1−s(x∧1y−∼) = 1−s(x∧1y−∼)+s(y−∼) = s((y−∼∗x)◦y−∼)−s(x∧1y−∼)+s(y−∼) = s(y−∼ ∗ x). Similarly we can prove s(y ◦ x∼−) = s(y∼− ◦ x). (2) By (p4) we have s(y−∼ ∗ x) = s((1a ◦ (1a ∗ y)) ∗ x) = s((1a ∗ x) ◦ (1a ∗ y)) = s(x− ◦ y−). Moreover we have s(y−∼ ∗ x−∼) = s((1a ◦ (1a ∗ y)) ∗ ((1a ◦ (1a ∗ x))) = s((1a ◦ ((1a ◦ (1a ∗ x))) ◦ (1a ∗ y)) = s(x−∼− ◦ y−) = s(x− ◦ y−) by Proposition 8. Using (1) we can get s(y−∼ ∗ x) = s(x− ◦ y−) = s(y−∼ ∗ x−∼) = s(y ∗ x−∼). Similarly we have s(y∼− ◦ x) = s(x∼ ∗ y∼) = s(y∼− ◦ x∼−) = s(y ◦ x∼−). (3) By Proposition 5.4(4) we get s(y−∼ ∗x∼) = s(y−∼) + s((y−∼ ∗x∼) ◦ y−∼)− s(y−∼ ◦ (y−∼ ∗x∼)) = s(y) + 1− s(x∼ ∧1 y−∼) = s(y) + 1− s(x∼−∼ ∧1 y) = s(y) + 1− s(x∼ ∧1 y) = s(y ◦ x∼). Similarly we can get s(y∼− ◦ x−) = s(y ◦ x−). Proposition 21. Let A be a lbp-BCI algebra and s be a Bosbach state on A. Then for all a ∈M(A) and x, y ∈ V (a), s(y∗x) = 1−s(x∧1y)+s(y) and s(y◦x) = 1−s(x∧2y)+s(y). Proof. Let a ∈ M(A) and x, y ∈ V (a). Note that x ∧1 y ≤ x, y and x ∧2 y ≤ x, y. By 19(1), we have s(y∗x) = s(y∗(x∧1y)) = 1−s(x∧1y)+s(y) and s(y◦x) = s(y◦(x∧2y)) = 1− s(x ∧2 y) + s(y). The following results are important for our study. Proposition 22. Let A be a lbp-BCI algebra and s be a Bosbach state on A. Then for all a ∈M(A) and x, y ∈ V (a), we have (1) s(x ∧1 y) = s(x ∧2 y). (2) s(x ∗ y) = s(x ◦ y). Proof. (1) First we prove the equality for x ≤ y. By Propositions 19(2) and 9(2), we have s(x ∧1 y) = s(y ∧1 x) = s(x) and s(x ∧2 y) = s(y ∧2 x) = s(x), that is s(x ∧1 y) = s(x ∧2 y). Now assume that x and y are arbitrary elements of V (a), where a ∈ M(A). Using Propositions 19(2) again and first part of the proof, we have s(x∧1 y) = s(x∧1 (x∧1 y)) = s((x ∧1 y) ∧1 x) = s((x ∧1 y) ∧2 x) ≤ s(y ∧2 x) = s(x ∧2 y). Dually, we can prove s(x ∧2 y) ≤ s(x ∧1 y). Hence s(x ∧1 y) = s(x ∧2 y). X.L. Xin, Y.J. Li, Y.L. Fu / Eur. J. Pure Appl. Math, 10 (3) (2017), 455-472 466 (2) It follows from Proposition 21 and the first equation. Consider the real interval [0,1] of reals equipped with the Lukasiewicz implication → L defined by x→ L y = min{1− x+ y, 1}, for all x, y ∈ [0, 1]. Definition 8. Let A be a lbp-BCI algebra. A state-morphism on A is a function m : A→ [0, 1] such that: (SM1) m(a) = 0,m(1a) = 1 for all a ∈M(A). (SM2) m(y ∗ x) = m(y ◦ x) = m(x)→ L m(y), for all x, y ∈ A. Proposition 23. Let A be a lbp-BCI algebra. Then every state-morphism on A is a Bosbach state on A. Proof. It is similar to the proof of [[4], Proposition 3.9]. Proposition 24. Let A be a lbp-BCI algebra. A Bosbach state m on A is a state-morphism if and only if m(x∧1 y) = min{m(x),m(y)} for all x, y ∈ A, or equivalently, m(x∧2 y) = min{m(x),m(y)} for all x, y ∈ A. Proof. It is similar to the proof of [[4], Proposition 3.10]. Let A be a lbp-BCI algebra and s be a Bosbach state on A. Define a set Ker(s) := {x ∈ A | s(x) = 1}. Ker(s) is called the kernel of s on A. Definition 9. Let A be a pseudo BCI algebra and I be a nonempty subset of A. If I satisfies the following conditions: (1) 0 ∈ I, (2) x ∈ I and y ∗ x ∈ I (or y ◦ x ∈ I) imply y ∈ I for all x, y ∈ A, I is called a pseudo ideal of A, simply called an ideal of A. Let I be a pseudo ideal of a pseudo BCI algebra A. If I satisfies 0∗x ∈ I and 0◦x ∈ I, we call I a closed pseudo ideal of A. If I satisfies x ∗ y ∈ I if and only if x ◦ y ∈ I, we call I a normal pseudo ideal of A. If I satisfies x ∗ y ∈ I if and only if x ◦ y ∈ I for all a ∈M(A), x, y ∈ V (a), we call I a local normal pseudo ideal of A. Proposition 25. Let A be a lbp-BCI algebra and s be a Bosbach state on A. Then Ker(s) is a closed and local normal proper ideal of A. Proof. Obviously, 0 ∈ Ker(s) and 1 /∈ Ker(s). Assume that x, y ∗ x ∈ Ker(s). Then we have 1 = s(x) and s(y ∗ x) = 1. It follows from Definition 5.1 that s(y) = s(x) + s(y ∗ x) − s(x ∗ y) = 2 − s(x ∗ y) ≥ 1 and thus s(y) = 1. Hence y ∈ Ker(s). This shows that Ker(s) is a proper ideal of A. For any x ∈ A, we have 0 ∗ x ∈ M(A) and 0 ∗ x ∈ M(A) by Corollary 2. Hence s(0 ∗ x) = 1 and s(0 ◦ x) = 1. It follows that 0 ∗ x ∈ Ker(s) and 0 ◦ x ∈ Ker(s). This shows that I is a closed pseudo ideal of A. By Proposition 22, we can get that A is local normal. X.L. Xin, Y.J. Li, Y.L. Fu / Eur. J. Pure Appl. Math, 10 (3) (2017), 455-472 467 Theorem 2. Let A be a pseudo BCI algebra and I be a pseudo ideal of A. Define a binary relation ”∼” on A by x ∼ y if and only if x∗y, y ∗x ∈ I if and only if x◦y, y ◦x ∈ I. Then ∼ is a congruence relation on A. Denote Cx = {y ∈ A | x ∼ y}. Define Cx ∗ Cy = Cx∗y and Cx ◦ Cy = Cx◦y. Denote A/I = {Cx | x ∈ A}. Then (A/I, ∗, ◦, C0) is a pseudo BCI algebra. If I is a closed pseudo ideal of A, then C0 = I. Proof. Obviously ∼ is reflexive and symmetric. Now we prove that it is transitive. Let x ∼ y and y ∼ z. Then x ∗ y, y ∗ z ∈ I. By (I1), (x ∗ z) ◦ (x ∗ y) ≤ y ∗ z, thus x ∗ z ∈ I. Similarly we can prove z ∗ x ∈ I. This shows that x ∼ z and hence ∼ is transitive. Thus it is an equivalent relation on A. We also can show that ∼ is a congruence relation on A and omit it. Denote A/I = {Cx|x ∈ A}. Then binary operations ”∗” and ”◦” on A/I are well-defined. Moreover we can show that (A/I, ∗, ◦) satisfies I1 − I5 in Definition 3.1. It follows that (A/I, ∗, ◦, C0) is a pseudo BCI algebra. Finally we assume that I is a closed pseudo ideal of A. Then for x ∈ I, we have 0 ∗ x ∈ I and x ∗ 0 = x ∈ I. Hence x ∼ 0, that is, x ∈ C0. Therefore C0 = I. Proposition 26. Let s be a Bosbach state on a lbp-BCI algebra A and K = ker(s). Then we have the following. (1) x/K ≤ y/K iff s(x∗y) = 1 iff s(x◦y) = 1, where x/K = {y ∈ A|y ∼ x} for all x ∈ A. (2) For all a ∈ M(A) and all x, y ∈ V (a), we have that x/K ≤ y/K iff s(y ∧1 x) = s(x) iff s(y ∧2 x) = s(x). (3) x/K = y/K iff s(x ∗ y) = s(y ∗ x) = 1 iff s(x ◦ y) = s(y ◦ x) = 1. (4) For all a ∈ M(A) and all x, y ∈ V (a), x/K = y/K iff s(x) = s(y) = s(x ∧1 y) iff s(x) = s(y) = s(x ∧2 y). (5) (A/K,≤, ∗, ◦, 0/K, 10/K) is a bounded pseudo-BCK algebra where 10 is the unit of V (0). (6) The mapping s̃ : A/K → [0, 1] defined by s̃(x/K) := s(x)(x ∈ A) is a Bosbach state on A/K. Proof. (1) By Theorem 2, we know that (A/K,≤, ∗, ◦, 0/K) is a pseudo-BCI algebra. Note that x/K ≤ y/K iff x/K ∗ y/K = (x ∗ y)/K = 0/K iff x ∗ y ∈ K iff s(x ∗ y) = 1. Similarly, x/K ≤ y/K iff x/K ◦ y/K = (x ◦ y)/K = 0/K iff x ◦ y ∈ K iff s(x ◦ y) = 1. (2) Let a ∈M(A) and x, y ∈ V (a). As s(x∗y) = 1−s(y∧1x)+s(x) by Proposition 21, we get x/K ≤ y/K iff s(y∧1x) = s(x). Similarly, we have x/K ≤ y/K iff s(y∧2x) = s(x). (3) It follows easily from (1). (4) It follows easily from (2). (5) First we prove M(A/K) = {0/K}. Let x/K ≤ 0/K. By (1), s(x ∗ 0) = 1. Note that 0∗x ∈M(A), then we have s(0∗x) = 1. By (3), x/K = 0/K. Thus 0/K ∈M(A/K). Conversely let x/K ∈ M(A/K). Obviously (0 ∗ (0 ∗ x))/K ≤ x/K. Hence (0 ∗ (0 ∗ x))/K = x/K. Since for any a ∈ M(A), s(a ∗ 0) = s(0 ∗ a) = 1, we have 0/K = a/K. Thus x/K = (0 ∗ (0 ∗ x))/K = 0/K. This shows that M(A/K) = {0/K}, and hence (A/K,≤, ∗, ◦, 0/K) is a pseudo-BCK algebra. Now we prove that 10/K is the greatest element of A/K. First we claim 10/K = 1a/K for all a ∈ M(A). Note that s(10) + s(1a ∗ 10) = s(1a) + s(10 ∗ 1a) and s(10) = s(1a) = 0 X.L. Xin, Y.J. Li, Y.L. Fu / Eur. J. Pure Appl. Math, 10 (3) (2017), 455-472 468 by Definition 7, we have s(1a ∗ 10) = s(10 ∗ 1a). Moreover s(1a ∗ 10) + s(a ◦ (1a ∗ 10)) = s(a) + s((1a ∗ 10) ◦ a) by Definition 7. By Corollary 1, a ◦ (1a ∗ 10) ∈ M(A), and so s(a◦(1a ∗10)) = 1. Since (1a ∗10)◦a = (1a ◦a)∗10 and 1a ◦a ∈ V (0) by Proposition 12, we have s((1a∗10)◦a) = s((1a◦a)∗10) = s(0) = 1. Hence s(1a∗10) = 1. By (3), 10/K = 1a/K for all a ∈ M(A). Let x/K ∈ A/K. Then x/K ≤ 1(0∗(0◦x))/K = 10/K. This shows that 10/K is the greatest element of A/K. It follows that (A/K,≤, ∗, ◦, 0/K, 10/K) is a bounded pseudo BCK algebra. (6) The fact that s̃ is a well-defined Bosbach state on A/K is now straightforward. Definition 10. Let A be a lbp-BCI algebra. Then (1) A is called good if x−∼ = x∼− for all x ∈ A. (2) A is with the condition (pDN) if x−∼ = x∼− = x for all x ∈ A. Proposition 27. Let s be a Bosbach state on a bounded pseudo-BCI algebra A and let K = ker(s). For every element x ∈ A, we have x−∼/K = x/K = x∼−/K, that is, A/K satisfies the (pDN) condition. Proof. It is similar to the proof of [[4], Proposition 3.14]. Remark 2. Let s be a Bosbach state on a pseudo-BCI algebra A. According to the proof of Proposition 27, we have s(x ∗ x−∼) = 1 = s(x ∗ x∼−) and s(x ◦ x−∼) = 1 = s(x ◦ x∼−). Theorem 3. Let A be a lbp-BCI algebra, s be a Bosbach state on A and K = ker(s). Then A/K is ∧1-commutative as well as ∧2-commutative. In addition, A/K is a ∧-semilattice and good. Proof. It is similar to the proof of [[4], Proposition 3.16]. Proposition 28. ([4]) Let A be a good pseudo-BCK algebra. We define a binary operation ⊗ on A by x⊗ y := y−∼ ∗ x∼. For all x, y ∈ A, the following hold: (1) x⊗ y = x∼− ◦ y−. (2) x⊗ y ≤ x, y. (3) x⊗ 1 = 1⊗ x = x∼−. (4) x⊗ 0 = 0⊗ x = 0. (5) (x⊗ y)−∼ = x⊗ y = x−∼ ⊗ y−∼. (6) ⊗ is associative. An MV-algebra is an algebra (A,⊕,− , 0) of type (2, 1, 0) such that (i) ⊕ is commutative and associative, (ii) x ⊕ 0 = x, (iii) x ⊕ 0− = 0−,(iv) x−− = x,(v) y ⊕ (y ⊕ x−)− = x ⊕ (x ⊕ y−)−. If we define x ∗ y = x ◦ y = y− ⊕ x, then (A, ∗, ◦, 1, 0) is a bounded pseudo-BCK algebra. An MV-state on an MV-algebra A is a mapping s : A→ [0, 1] such that s(1) = 1 and s(a⊕b) = s(a)+s(b) whenever a�b = 0. Every MV-algebra admits at least one MV-state, and due to [17], every MV-state on A coincides with a Bosbach state on the BCK algebra A and vice versa. X.L. Xin, Y.J. Li, Y.L. Fu / Eur. J. Pure Appl. Math, 10 (3) (2017), 455-472 469 We note that the radical, Rad(A), of an MV-algebra A is the intersection of all maximal ideals of A([7]). Proposition 29. ([9]). In any MV-algebra A the following conditions are equivalent: (a) Rad(A) = 0. (b) nx ≤ x− for all n ∈ N implies x = 0. (c) nx ≤ y− for all n ∈ N implies x ∧ y = 0. (d) nx ≤ y for all n ∈ N implies x�y = x, where nx = x1⊕· · ·xn with x1 = · · · = xn = x. Remark 3. An MV-algebra A is archimedean in the sense of [9] if it satisfies the condition (b) of Proposition 29 and A is archimedean in Belluces sense [1] if it satisfies the condition (d) of Proposition 29. By Proposition 29 the two definitions of archimedean MV-algebras are equivalent. Theorem 4. Let s be a Bosbach state on a lbp-BCI algebra A and let K = Ker(s). Then (A/K,⊕,− , 0/K), where a/K ⊕ b/K = (b ∗ a−)/K and (a/K)− = a−/K, is an archimedean MV-algebra and the map ŝ(a/K) := s(a) is an MV-state on this MV-algebra. Proof. It is similar to the proof of [[4], Theorem 3.20]. By Theorem 3, A/K is a good pseudo-BCK algebra that is a ∧-semilattice and s̃ on A/K is a Bosbach state such that Ker(s̃) = {0/K}. Due to [[20], Proposition 3.4.7], (A/K)/Ker(s̃) is term-equivalent to an MV-algebra that is archimedean and s̃ is an MV- state on it. Since A/K = (A/K)/Ker(s̃), the same is true also for A/K, and this proves the theorem. In the following, we give properties of state-morphisms on lbp-BCI algebras. Lemma 1. Let A be a lbp-BCI algebra and m be a state-morphism on A. Then we have the following. (1) m(y−∼ ∗ x∼) = min{m(x) +m(y), 1}, for all a ∈M(A) and x, y ∈ V (a). (2) m(x∼− ◦ y−) = min{m(x) +m(y), 1}, for all a ∈M(A) and x, y ∈ V (a). Proof. Assume that m is a state-morphism on A, so it is a Bosbach state on A. By Propositions 19 and 20, for for all a ∈ M(A) and x, y ∈ V (a), we have m(y−∼ ∗ x∼) = m(y ∗ x∼) = m(x∼) → L m(y) = m(x)∼ → L m(y) = min{1 − m(x)∼ + m(y), 1} = min{m(x) + m(y), 1}. Similarly we can prove m(x∼− ◦ y−) = min{m(x) + m(y), 1}, for all a ∈M(A) and x, y ∈ V (a). Proposition 30. Let A be a lbp-BCI algebra and s be a Bosbach state on A. Then the following are equivalent: (1) s is a state-morphism. (2) ker(s) is a maximal ideal of A. Proof. It is similar to the proof of [[4], Proposition 3.22]. X.L. Xin, Y.J. Li, Y.L. Fu / Eur. J. Pure Appl. Math, 10 (3) (2017), 455-472 470 Lemma 2. Let m be a state-morphism on a lbp-BCI algebra A and K = ker(m). Then (1) a/K ≤ b/K if and only if m(a) ≤ m(b), (2) a/K = b/K if and only if m(a) = m(b). Proof. It is similar to the proof of [[4], Lemma 3.23]. Proposition 31. Let A be a lbp-BCI algebra and m1,m2 be two state-morphisms on A such that ker(m1) = ker(m2). Then m1 = m2. Proof. By Proposition 23, m1 and m1 are two Bosbach states on A. Since ker(m1) = ker(m2), we have A/ker(m1) = A/ker(m2). By the proof of Proposition 30, we have that A/ker(m1) is in fact an MV-subalgebra of the MV-algebra of the real interval [0, 1]. But ker(m̂1) = 0/K = ker(m̂2). Hence, by [[11], Proposition 4.5], m̂1 = m̂2, consequently, m1 = m2. Let A be a lbp-BCI algebra. We say that a Bosbach state s is extremal if for any 0 < λ < 1 and for any two Bosbach states s1, s2 on A, s = λs1 + (1−λ)s2 implies s1 = s2. Summarizing previous characterizations of state-morphisms, we have the following result. Theorem 5. Let s be a Bosbach state on a lbp-BCI algebra A. Then the following are equivalent: (1) s is an extremal Bosbach state. (2) s(x ∧1 y) = max{s(x), s(y)} for all x, y ∈ A. (3) s(x ∧2 y) = max{s(x), s(y)} for all x, y ∈ A. (4) s is a state-morphism. (5) ker(s) is a maximal ideal. Proof. It is similar to the proof of [[4], Theorem 3.26]. 5. Conclusions Until now, the states on unbounded algebraic structures have been studied for Hilbert algebras and integral residuated lattices in [2] and [6], respectively. In this paper, we first study state theory on non-bounded algebraic structures, and in- troduce a notion of state on pseudo-BCI algebras. In order to adapt a state to pseudo-BCI algebras, we first discuss the structure of pseudo-BCI algebras, which can be decomposed in to the union of it’s branches. Note that for all a ∈ M(A) and a 6= 0, V (a) is not a BCK-algebra, hence the structure of pseudo-BCI algebras is different from the structure of pseudo-BCK algebras. Therefore it is valuable to study state theory on pseudo-BCI alge- bras. Moreover we introduce a notion of local bounded pseudo-BCI algebras and set up the theory of states on such algebraic structure. We also introduce a notion of state-morphisms on local bounded pseudo-BCI algebras and discuss the relations between Bosbach states and state-morphisms. By use of state’s theory, we discuss the relation between pseudo- BCI algebras and MV-algebras. In the next work, we will consider the following problem: satisfying what apposite conditions a local bounded pseudo-BCI algebra admits a Bosbach state? REFERENCES 471 Acknowledgements This research is partially supported by a grant of National Natural Science Foundation of China (11571281,61602359), China Postdoctoral Science Foundation (2015M582618), China 111 Project (B16037). 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