Ratio Mathematica Volume 41, 2021, pp. 79-100 On extended quasi-MV algebras Mengmeng Liu* Hongxing Liu† Abstract In this paper, we introduce a new algebraic structure called extended quasi-MV algebras, which are generalizations of quasi-MV algebras. The notions of ideals, ideal congruences and filters in Equasi-MV algebras were introduced and their mutual relationships were investi- gated. There is a bijection between the set of all ideals and the set of all ideal congruences on an Equasi-MV algebra. Keywords: Equasi-MV algebras; Quasi-MV algebras; Idempotent elements; Ideal congruences; Filters 2020 AMS subject classifications: 03G27 1 *School of Mathematics and Statistics, Shandong Normal University, 250014, Jinan, P. R. China; 535236069@qq.com. †School of Mathematics and Statistics, Shandong Normal University, 250014, Jinan, P. R. China; lhxshanda@163.com. 1Received on October 15, 2021. Accepted on December 18, 2021. Published on December 31, 2021. doi: 10.23755/rm.v41i0.680. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 79 Mengmeng Liu, Hongxing Liu 1 Introduction MV-algebras were introduced by Chang Chang [1958] as an algebraic counter- part of infinite valued logic. There are many papers on MV-algebras. Also, many algebraic structures are defined, which extend the notion of MV-algebras. Quan- tum computation logics M. L. Dalla Chiara and Leporini [2005] received more attention in recent years, which are new forms of quantum logics G. Cattaneo and Leporini [2004]. These logics determine the meaning of a sentence with a mixture of quregisters M. L. Dalla Chiara and Greechie [2013]. Corresponding to quan- tum computational, Ledda, Konig, Paoli and Giuntini introduced the notion of quasi-MV algebras in A. Ledda and Giuntini [2006], which are generalizations of MV-algebras. The element 0 in a quasi-MV algebra is not necessarily a neutral el- ement of the operation ⊕. Since then, many authors continued to study quasi-MV algebras. For example, Ledda etc. studied some properties of quasi-MV alge- bras and √ ′ quasi-MV algebras F. Bou and Freytes [2008], F. Paoli and Freytes [2009]; Chen introduced pseudo-quasi-MV algebras which are non-commutative generalizations of quasi-MV algebras Liu and Chen [2016]. EMV-algebras (extended MV-algebras) Dvurečenskij and Zahiri [2019] are also generalizations of MV-algebras. An EMV-algebra does not necessarily have a top element. Dvurečenskij and Zahiri gave some properties of EMV-algebras. The notions of ideals, congruences and filters in EMV-algebras were also intro- duced and the relationships between them were investigated. One of the main results is that every EMV-algebra can be embedded into an EMV-algebra with a top element. Liu presented EBL-algebras in Liu [2020], which extended the no- tion of BL-algebras. The author gave some properties of EBL-algebras. Also, the concepts of ideals, congruences and filters were introduced and the relationships between them were studied. Inspired by Dvurečenskij and Zahiri [2019], we shall give the definition of Equasi-MV algebras. In these algebras, 0 is not necessarily the neutral element and the complement element of 0 does not necessarily exist. The structure of this paper is as follows. In Sect.2, we give some definitions and results of quasi-MV algebras. In Sect.3, we introduce Equasi-MV algebras and present some examples of Equasi-MV algebras. In Sect.4, we define ideals and ideal congruences in Equasi-MV algebras. And we study the relationships between them. In Sect.5, we introduce the notions of filters and prime ideals. Moreover, every Equasi-MV algebra has at least one maximal ideal. 80 On extended quasi-MV algebras 2 Preliminaries In this section, we will give some notions and results on quasi-MV algebras, which will be used in the following. A quasi-MV algebra A. Ledda and Giuntini [2006] is an algebra A=〈A,⊕,′,0,1〉 of type 〈2,1,0,0〉 satisfying the following conditions: QMV1) x⊕ (y ⊕z) = (x⊕z)⊕y; QMV2) x′′ = x; QMV3) x⊕1 = 1; QMV4) (x′ ⊕y)′ ⊕y = (y′ ⊕x)′ ⊕x; QMV5) (x⊕0)′ = x′ ⊕0; QMV6) (x⊕y)⊕0 = x⊕y; QMV7) 0′ = 1. In any quasi-MV algebra A, we can define the following operations: x⊗y = (x′ ⊕y′)′; x d y = x⊕ (x′ ⊗y); x e y = x⊗ (x′ ⊕y). It is obvious that x d y = (x d y)⊕0 and x e y = (x e y)⊕0. Moreover, we can also define an binary relation 6 on A as follows: x 6 y iff x e y = x ⊕ 0. The relation 6 is a preordering of A, but not a partial ordering. Lemma 2.1. [A. Ledda and Giuntini, 2006, Lemma 8] Let A be a quasi-MV algebra. For all x,y,z ∈ A, the following statements are equivalent. (i) x 6 y; (ii) x′ ⊕y = 1; (iii) x d y = y ⊕0. In the following, we give some properties of quasi-MV algebras, including a few properties of preordering 6 and the operations e and d. Lemma 2.2. [A. Ledda and Giuntini, 2006, Lemma 11] Let A be a quasi-MV algebra. For all x,y,z,w ∈ A: (i) x⊕0 6 y⊕0, y⊕0 6 x⊕0 imply x⊕0 = y⊕0; (vi) x 6 x⊕0 and x⊕0 6 x; (ii) x 6 y and z 6 w imply x⊕z 6 y⊕w; (vii) x⊗y 6 z iff x 6 y′⊕z; (iii) x 6 y and z 6 w imply x⊗z 6 y⊗w; (viii) if x 6 y, then y′ 6 x′; (iv) x 6 y and z 6 w imply xez 6y ew; (ix) 0 6 x, x 6 1. (v) x 6 y and z 6 w imply xdz 6 ydw; Lemma 2.3. [A. Ledda and Giuntini, 2006, Lemma 12] Let A be a quasi-MV 81 Mengmeng Liu, Hongxing Liu algebra. For all x,y,z ∈ A: (i) xey = yex; (vii) x⊗(ydz) = (x⊗y)d(x⊗z); (ii) xdy = ydx; (viii) xe(yez) = (xey)ez; (iii) xey 6 x, y and x, y 6 xdy; (ix) xd(ydz) = (xdy)dz; (iv) if x 6 y, z, then x 6y ez; (x) x 6 xex and xex 6 x; (v) if x, y 6 z, then xdy 6 z; (xi) (xey)′ = x′dy′ and (xdy)′ = x′ey′. (vi) x⊕(yez) = (x⊕y)e(x⊕z); The following lemma gives the distributivity between e and d on quasi-MV algebras. Lemma 2.4. Let A be a quasi-MV algebra. For all x,y,z ∈ A, (i) (x d y) e z = (x e z) d (y e z); (ii) (x e y) d z = (x d z) e (y d z); (iii) x e (y ⊕z) 6 (x e y)⊕ (x e z); (iv) (x d y)⊗ (x d z) 6 x d (y ⊗z). Proof. (i) For any x,y ∈ A, we have x,y 6 xdy and so xez, yez 6 (xdy)ez by Lemma 2.2 (iv). It follows from Lemma 2.3 (v) that (xez)d(yez)6(xdy)ez. Conversely, we have (x d y) e z = (x d y)⊗ ((x d y)′ ⊕z) = (x d y)⊗ ((x′ ⊕z) e (y′ ⊕z)) (Lemma 2.3 (xi) and (vi)) 6 (x⊗ (x′ ⊕z)) d (y ⊗ (y′ ⊕z)) (Lemma 2.3 (vii) and (iii)) = (x e z) d (y e z). Then ((x e z) d (y e z)) ⊕ 0 6 ((x d y) e z) ⊕ 0 and ((x d y) e z) ⊕ 0 6 ((x e z) d (y e z))⊕0. Note that ((x e z) d (y e z))⊕0 = (x e z) d (y e z) and ((x d y) e z) ⊕ 0 = (x d y) e z. It follows that (x e z) d (y e z) = (x d y) e z by Lemma 2.2 (i). Similarly, we can prove (ii). (iii) For any x,y,z ∈ A, since x 6 x⊕0 6 x⊕y, we have (x e y)⊕ (x e z) = ((x e y)⊕x) e ((x e y)⊕z) (Lemma 2.3 (vi)) = (x⊕x) e (y ⊕x) e (x⊕z) e (y ⊕z) > x e x e x e (y ⊕z) = (x⊕0) e x e (y ⊕z) (Lemma 2.3 (x)) = (x⊕0) e (y ⊕z). Note that (x⊕0)e(y⊕z) = xe(y⊕z). It follows that xe(y⊕z) 6 (xey)⊕(xez). (iv) For any x,y,z ∈ A, it follows from (x⊗y)′ ⊕y = x′ ⊕y′ ⊕y = 1 that 82 On extended quasi-MV algebras x⊗y 6 y. Then we have (x d y)⊗ (x d z) = ((x d y)⊗x) d ((x d y)⊗z) (Lemma 2.3 (vii)) = (x⊗x) d (y ⊗x) d (x⊗z) d (y ⊗z) 6 x d x d x d (y ⊗z) = (x⊕0) d x d (y ⊗z) (Lemma 2.3 (x)) = (x⊕0) d (y ⊗z). Note that (x ⊕ 0) d (y ⊗ z) = x d (y ⊗ z). It follows that (x d y) ⊗ (x d z) 6 x d (y ⊗z).2 Let A be a quasi-MV algebra and a ∈ A. If a ⊕ a = a, we call a to be idempotent. We use I(A) to denote the set of all idempotent elements of A. For a ∈ A, we call a regular if a⊕0 = a. We denote the set of all regular elements of A by R(A). Lemma 2.5. Let A be a quasi-MV algebra. For any x ∈ A, a ∈I(A), we have (i) x⊕a = x d a; (ii) x⊗a = x e a. Proof. (i) For any x ∈ A and a ∈ I(A), we have x,a 6 x ⊕ a. Then x d a 6 x⊕a by Lemma 2.3 (v). Conversely, (x⊕a)⊗ (x d a)′ = (x⊕a)⊗ (x′ e a′) (Lemma 2.3 (xi)) 6 ((x⊕a)⊗x′) e ((x⊕a)⊗a′) (Lemma 2.2(iii) and 2.3(iv)) = (a e x′) e (x e a′) = (a e a′) e (x e x′) = 0 e (x e x′) = 0. This means that (x⊕a)′ ⊕ (x d a) = 1. It follows that x⊕a 6 x d a. (ii) By (i), we have x′ ⊕a′=x′ d a′, that is (x′ ⊕a′)′ = (x′ d a′)′ = x e a. It follows that x e a = x⊗a.2 The application of the above lemma will be reflected in the following proof process. Example 2.1. [A. Ledda and Giuntini, 2006, Example 3] The Diamond is the 4- element quasi-MV algebra, where the operations ⊕ and ′ are defined as following tables: ⊕ 0 a b 1 0 0 b b 1 a b 1 1 1 b b 1 1 1 1 1 1 1 1 ′ 0 1 a a b b 1 0 Remark that a⊕a = 1, but aea = (a′⊕(a′⊕a)′)′ = (a⊕(a⊕a)′)′ = b 6= 1. 83 Mengmeng Liu, Hongxing Liu 3 Equasi-MV algebras In the section, we shall define the notion of extended quasi-MV algebras, which are generalizations of quasi-MV algebras. Some basic properties of these algebras are presented. Definition 3.1. A extended quasi-MV algebra (abbreviated as Equasi-MV alge- bra) is an algebra A=〈A,⊕,0〉, if the following conditions are satisfied: EQMV1) 〈A,⊕,0〉 is a commutative preordered semigroup and (x⊕y)⊕0 = x⊕y for all x,y ∈ A; EQMV2) for each x ∈ A, there is b ∈I(A) such that x 6 b, and the element λb(x) = min{z ∈ [0,b] : z ⊕x = b} exists in A for all x ∈ [0,b] such that 〈[0,b],⊕,λb,0,b〉 is a quasi-MV algebra. Note that for any x,y ∈ A, there exist a,b ∈I(A) such that x 6 a and y 6 b. Then there exists c ∈I(A) such that a,b 6 c. In fact, take c = a⊕b. It is obvious that a,b 6 a⊕b and a⊕b ∈I(A). Therefore, an Equasi-MV algebra has enough idempotent elements. That is, for all x ∈ A, there is a ∈I(A) such that x 6 a. Let A be an Equasi-MV algebra. For all n ∈ N and x ∈ A, we define 0.x = 0, 1.x = x, · · · , (n + 1).x = n.x⊕x. An Equasi-MV algebra 〈A,⊕,0〉 is called a proper Equasi-MV algebra if 0 has no complement element. Example 3.1. If 〈A,⊕,′ ,0,1〉 is a quasi-MV algebra, then 〈A,⊕,0〉 is an Equasi- MV algebra. Also, if 〈A,∨,∧,⊕,0〉 is an EMV-algebra, it is obvious that 〈A,⊕,0〉 is an Equasi-MV algebra. Example 3.2. Let 〈A,⊕,′ ,0,1〉 be a quasi-MV algebra and 〈B,∨,∧,⊕,0〉 be an EMV-algebra. We define that the operation on the algebra A×B is point by point. That is, for any 〈x1,x2〉,〈y1,y2〉 ∈ A×B, 〈x1,x2〉⊕〈y1,y2〉 = 〈x1 ⊕y1,x2 ⊕y2〉. And the least element of A×B is 0 = 〈0,0〉. For any x ∈ B, there exists b ∈I(B) such that x 6 b. Then for any 〈x1,x2〉 ∈ A × B, there exists 〈1,b〉 ∈ I(A) × I(B). It suffices to show that 〈[〈0,0〉,〈1,b〉],⊕,λ〈1,b〉,〈0,0〉,〈1,b〉〉 is a quasi-MV algebra. We define λ〈1,b〉(〈x1,x2〉) = 〈(x1)′,λb(x2)〉, for all 〈x1,x2〉 ∈ A×B. As a result, A×B is an Equasi-MV algebra. Example 3.3. Let e be a smallest idempotent of an Equasi-MV algebra A. Then an Equasi-MV algebra is the algebra S=〈A×A,⊕S,0S〉, where: (i) 0S = 〈0, e 2 〉; (ii) xS ⊕S yS = 〈x1 ⊕y1, e2〉, for all x S = 〈x1,x2〉 and yS = 〈y1,y2〉. For any a ∈ I(A), we define aS = 〈a, e 2 〉. Then aS = aS ⊕aS ∈ I(S). Now we show that 〈[0S,aS],⊕S,λaS,0S,aS〉 is a quasi-MV algebra, where λaS(xS) = 84 On extended quasi-MV algebras 〈λa(x1),x2〉 and a ∈ I(A). It is easy to show that λaS(xS) is the least element such that xS ⊕zS = aS for all xS ∈ [0S,aS]. It is clear that λaSλaS(x S) = λaS〈λa(x1),x2〉 = 〈x1,x2〉 = xS. And λaS(xS⊕S 0S) = λaS〈x1 ⊕ 0, e2〉 = 〈λa(x1) ⊕ 0, e 2 〉, λaS(xS) ⊕ 0S = 〈λa(x1),x2〉⊕ 0S = 〈λa(x1)⊕0, e2〉. What’s more, λaS(0 S) = 〈λa(0), e2〉 = 〈a, e 2 〉 = aS. Example 3.4. Let 〈A,∨,∧,0〉 be a generalized Boolean algebra Conrad and Dar- nel [1997]. For any x,y ∈ [0,b], where ⊕ = ∨ and λb(x) is the unique relative complement of x in [0,b]. Then 〈A,⊕,0〉 is an EMV-algebra by Example 3.2 (2) in Dvurečenskij and Zahiri [2019]. Hence, 〈A,⊕,0〉 is an Equasi-MV algebra. Example 3.5. Let 〈A,⊕,′ ,0,1〉 be a quasi-MV algebra and 〈B,∨,∧,0〉 be a gen- eralized Boolean algebra. It is easy to show that A×B is an Equasi-MV algebra. Proof. The operation ⊕ on A × B is defined pointwise. For all 〈x,y〉 ∈ A × B, there exist a ∈ I(A) and b ∈ I(B) such that 〈x,y〉 6 〈a,b〉 and 〈[〈0,0〉,〈a,b〉],⊕,λ〈a,b〉,〈0,0〉,〈a,b〉〉 is a quasi-MV algebra. Let’s give a specific description of the above example. Let the Diamond (Ex- ample 2.6) be the 4-element quasi-MV algebra A and M = 〈M,∨,∧,0〉 be the generalized Boolean algebra Conrad and Darnel [1997], where M is the set of components of any positive element N+ and the least element 0 := ∅. That is, M = {N : N ⊆ N+}. Then every element N in M is idempotent. It is easily shown that A×M with the pointwise operation is an Equasi-MV algebra. 2 Example 3.6. Let S = 〈[0,1] × [0,1],⊕,′ ,0,1〉 be a standard quasi-MV algebra A. Ledda and Giuntini [2006, Example 5]. Let A = S⊕S⊕S⊕··· . Then A is an Equasi-MV algebra. Proof. Obviously, 〈A,⊕,0〉 is a commutative preordered semigroup and (x⊕ y) ⊕ 0 = x⊕y for all x,y ∈ A. For any x,y ∈ A. Suppose x = (xi), y = (yi). If xi 6= 0 or yi 6= 0, there exists ui ∈ I(A) such that xi,yi 6 ui for all i > 1. If xi = yi = 0, take ui = 0. We have an idempotent u = (ui) ∈ A such that x,y 6 u and 〈[0,u],⊕,λu,0,u〉 is a quasi-MV algebra. 2 Remark 3.1. Let A be an Equasi-MV algebra. For all x,y ∈ A, there exists b ∈ I(A) such that x,y ∈ [0,b]. In the quasi-MV algebra 〈[0,b],⊕,λb,0,b〉, we denote x db y = λb(λb(x)⊕y)⊕y, x eb y = λb(λb(x)⊕λb(λb(x)⊕y)). Proposition 3.1. Let A be an Equasi-MV algebra and a,b ∈ I(A) such that a 6 b. For each x ∈ [0,a], we have (i) λb(a) is an idempotent, and λa(a)=0; (ii) λa(x)⊕0 = λb(x) e a; (iii) λb(x)⊕0 = λa(x)⊕λb(a); (iv) λa(x) 6 λb(x). 85 Mengmeng Liu, Hongxing Liu Proof. Since 〈[0,b],⊕,λb,0,b〉 is a quasi-MV algebra and a ∈ I(A), by Lemma 2.5 (i) we get that x⊕a = x d a for all x ∈ [0,b]. (i) Since 〈[0,b],⊕,λb,0,b〉 is a quasi-MV algebra, λb(a) is also an idempotent element by Lemma 26 in A. Ledda and Giuntini [2006]. It is obvious λa(a) = 0 in the quasi-MV algebra 〈[0,a],⊕,λa,0,a〉. (ii) For all x ∈ [0,a], we have (λb(x) e a)⊕ (x⊕0) = (λb(x)⊕ (x⊕0)) e (a⊕ (x⊕0)) (Lemma 2.3 (vi)) = b e a = a. It follows that λa(x) ⊕ 0 = λa(x ⊕ 0) 6 λb(x) e a in the quasi-MV algebra 〈[0,b],⊕,λb,0,b〉. Conversely, since b = a ⊕ λb(a) = x ⊕ (λa(x) ⊕ λb(a)), we get λb(x) 6 λa(x) ⊕ λb(a). Since λb(a) is an idempotent, by Lemma 2.5 (i) we have λa(x)⊕λb(a) = λa(x) d λb(a). Hence, λb(x) 6 λa(x) d λb(a). Thus λb(x) e a 6 (λa(x) d λb(a)) e a (Lemma 2.2 (iv)) = λa(x)⊕0 (Lemma 2.4 (i)). Summary of the above results, we get that λa(x)⊕0 = λb(x) e a. (iii) By (ii) we have λa(x)⊕λb(a) = (λa(x)⊕0)⊕λb(a) = (λb(x) e a)⊕λb(a) = λb(x) d λb(a) (Lemma 2.3 (vi) and Lemma 2.5 (i)). It follows from x 6 a that λb(a) 6 λb(x). Then λb(x) d λb(a) = λb(x) ⊕ 0. Therefore, λb(x)⊕0 = λa(x)⊕λb(a). (iv) It follows from (ii) or (iii).2 The following statement shows that da and ea on [0,a] are coincide with d and e on A, respectively. Proposition 3.2. Let A be an Equasi-MV algebra. For all x,y ∈ A, there exist a,b ∈I(A) such that x,y ∈ [0,a] and x,y ∈ [0,b]. Then we have (i) x ea y = x eb y; (ii) x da y = x db y. Proof. (i) By Definition 3.1, for all a,b ∈ I(A), there exists c ∈ I(A) such that a,b 6 c. Then we have x dc y = x⊕λc(x⊕λc(y)⊕0) = x⊕λc(x⊕λa(y)⊕λc(a)) (Proposition 3.1 (iii)) = x⊕ (λc(x⊕λa(y))⊗c a) (the definition of ⊗c) = x⊕ (λc(x⊕λa(y)) e a) (Lemma 2.5 (ii)) = x⊕ ((λa(x⊕λa(y)) d λc(a)) e a) (Proposition 3.1(iii), Lemma 2.5(i)) = x⊕ (λa(x⊕λa(y)) e a) (Lemma 2.4 (i)) = x⊕ (λa(x⊕λa(y))⊕0) = x da y. 86 On extended quasi-MV algebras Similarly, we can show that x dc y = x db y. Hence, x da y = x db y. (ii) We also have xecy = λc(λc(x)⊕λc(λc(x)⊕y)) = λc(λc(x)⊕λc(λa(x)⊕y⊕λc(a))) (Proposition 3.1 (iii)) = λc(λc(x)⊕(λc(λa(x)⊕y)⊗ca)) (definition of ⊗c) = λc(λc(x)⊕((λa(λa(x)⊕y)⊕λc(a))⊗c a)) (Proposition 3.1 (iii)) = λc(λc(x)⊕((λa(λa(x)⊕y)⊕λc(a))ea)) (Lemma 2.5 (i)) = λc(λc(x)⊕(λa(λa(x)⊕y)ea)) ( Lemma 2.4 (i)) = λc(λc(x)⊕λa(λa(x)⊕y)) = λc(λa(x)⊕λa(λa(x)⊕y)⊕λc(a)) (Proposition 3.1 (iii)) = λc(λa(x)⊕λa(λa(x)⊕y))⊗ca (definition of ⊗c) = (λa(λa(x)⊕λa(λa(x)⊕y))⊕λc(a))⊗ca (Proposition 3.1 (iii)) = λa(λa(x)⊕λa(λa(x)⊕y))ea = xeay. Similarly, we can show that x ec y = x eb y and so x ea y = x eb y.2 Definition 3.2. Let A be an Equasi-MV algebra and x,y ∈ [0,a] where a ∈ I(A). A preordering 6a on the quasi-MV algebra 〈[0,a],⊕,λa,0,a〉 defined as follows: x 6a y ⇐⇒ x ea y = x⊕0. By Proposition 3.2, for any x,y 6 a,b, where a,b ∈ I(A), we have x 6a y ⇐⇒ x 6 y ⇐⇒ x 6b y. Then we can also define a preordering 6 on A by x 6 y ⇐⇒ x e y = x⊕0, where x e y = x ea y. Lemma 3.1. Let A be an Equasi-MV algebra. For all x,y ∈ A, the operation ⊗: A × A → A defined by x ⊗ y = λa(λa(x) ⊕ λa(y)), where a ∈ I(A) and x,y 6 a. Then (i) the well-defined binary operation ⊗ on A is not determined by the choice of a and is also order preserving and associative. (ii) if x,y ∈ A, x 6 y, then y⊗λa(x) = y⊗λb(x) and y⊕0 = x⊕(y⊗λa(x)) for all a,b ∈I(A) and x,y 6 a,b. (iii) if x,y ∈ [0,a] and a ∈ I(A), then x ⊗ λa(y) = x ⊗ λa(x e y) and x⊕0 = (x e y)⊕ (x⊗λa(y)). (iv) an element a ∈ A is idempotent iff a⊗a = a. Proof. (i) Let x,y ∈ A and a,b ∈ I(A) such that x,y 6 a,b. We claim that λa(λa(x)⊕λa(y)) = λb(λb(x)⊕λb(y)). Indeed, there exists an element c ∈I(A) 87 Mengmeng Liu, Hongxing Liu such that a,b 6 c. Then λc(λc(x)⊕λc(y)) = λc(λa(x)⊕λc(a)⊕λa(y)⊕λc(a)) (Proposition 3.1 (iii)) = λc(λa(x)⊕λa(y))⊗c λc(λc(a)) (Propsition 3.1 (i)) = λc(λa(x)⊕λa(y)) e a (Lemma 2.5 (ii)) = (λa(λa(x)⊕λa(y))⊕λc(a)) e a (Lemma 3.1 (iii)) = (λa(λa(x)⊕λa(y)) d λc(a)) e a (Lemma 2.5 (i)) = λa(λa(x)⊕λa(y)) e a = λa(λa(x)⊕λa(y)). Similarly, we have λc(λc(x)⊕λc(y)) = λb(λb(x)⊕λb(y)). Let x,y,z ∈ A. There exists c ∈ I(A) such that x,y,z 6 c. It follows from the definition of ⊗ that x⊗y, y ⊗z ∈ [0,c]. Then (x⊗y)⊗z = λc(λc(x⊗y)⊕λc(z)) = λc((λc(x)⊕λc(y))⊕λc(z)) = λc(λc(x)⊕ (λc(y)⊕λc(z))) = λc(λc(x)⊕λc(y ⊗z)) = x⊗ (y ⊗z). This proves that ⊗ is associative. It is easy to prove that ⊗ is order preserving. (ii) Let x 6 y and x,y 6 a,b, where a,b ∈I(A). There exists c ∈I(A) such that a,b 6 c. By Proposition 3.1, we have y ⊗λa(x) = λc(λc(y)⊕λc(λa(x))) = λc(λc(y)⊕λc(λa(x))⊕0) = λc(λc(y)⊕λc(λa(x)⊕0)) = y ⊗ (λa(x)⊕0). Then y ⊗λc(x) = y ⊗ (λc(x)⊕0) = y ⊗ (λa(x)⊕λc(a)) = y ⊗ (λa(x) d λc(a)) (Lemma 2.5 (i)) = (y ⊗λa(x)) d (y ⊗λc(a)) (Lemma 2.3 (vii)). Since λc(a) 6 λc(y), we have y⊗λc(a) 6 y⊗λc(y) = 0, where y 6 a 6 c. This implies y ⊗ λc(x) = y ⊗ λa(x). Similarly, we have y ⊗ λc(x) = y ⊗ λb(x). It follows that y ⊗λa(x) = y ⊗λb(x). In the quasi-MV algebra 〈[0,a],⊕,λa,0,a〉, we have x⊕ (y ⊗λa(x)) = x⊕λa(λa(y)⊕x) = x d y = y ⊕0. (iii) Let x,y 6 a and a ∈I(A). We have x⊗λa(x e y) = x⊗ (λa(x) d λa(y)) = (x⊗λa(x)) d (x⊗λa(y)) (Lemma 2.3 (vii)) = x⊗λa(y). 88 On extended quasi-MV algebras (x e y)⊕ (x⊗λa(y)) = (x e y)⊕ (x⊗λa(x e y)) = (x e y)⊕λa(λa(x)⊕ (x e y)) = x⊕λa(x⊕λa(x e y)) (QMV 4) = x⊕λa(x⊕λa(x)⊕λa(λa(x)⊕y)) = x⊕0. (iv) =⇒: Suppose a,b ∈I(A) with a 6 b. We have λb(a)⊕λb(a) = λb(a) by Proposition 3.1 (i). In the quasi-MV algebra 〈[0,b],⊕,λb,0,b〉, we have a⊗a = λb(λb(a)⊕λb(a)) = λb(λb(a)) = a. ⇐=: For each a ∈ A, there exists b ∈ I(A) such that a 6 b. Suppose a⊗a = a. We have λb(λb(a)⊕λb(a)) = a. Then λb(λb(λb(a)⊕λb(a))) = λb(a). It follows from λb(a) ⊕ λb(a) = λb(a) that λb(a) ∈ I(A). By Proposition 3.1 (i), we have λb(λb(a)) ⊕ λb(λb(a)) = λb(λb(a)). That is a ⊕ a = a. It implies a ∈I(A). 2 Theorem 3.1. Let A be an Equasi-MV algebra. Then 〈R(A),dR,eR,⊕R,0R〉 is an EMV-subalgebra of A. Proof. It is obvious that R(A) is closed under the operations dR,eR,⊕R,0R. For all x,y ∈R(A), there exists a ∈I(A) such that x,y 6 a. Then [0,a]∩R(A) is an MV-algebra of [0,a] by Lemma 15 in A. Ledda and Giuntini [2006]. This means that R(A) is an EMV-subalgebra of A. 2 4 Ideals and congruences In this section, we give the notions of ideals and ideal congruences of Equasi- MV algebras. We also give an equivalent definition of ideals. Moreover, there is a one-to-one correspondence between the set of all ideals and the set of all ideal congruences. Definition 4.1. Let A be an Equasi-MV algebra. An equivalence relation θ on A is called a congruence, if the following conditions hold: (i) θ is compatible with ⊕; (ii) for all b ∈ I(A),θ ∩ ([0,b] × [0,b]) is a congruence on the quasi-MV algebra 〈[0,b],⊕,λb,0,b〉. The set of all congruences on A represented by Con(A). Definition 4.2. Let A1,A2 be two Equasi-MV algebras. We call a map f : A1 −→ A2 to be an Equasi-MV homomorphism, if it satisfies the following state- ments: (i) f(x⊕y) = f(x)⊕f(y) and f(0) = 0, for all x,y ∈ A1; (ii) for all x,y ∈ [0,a] and a ∈I(A1), f(λa(x)) = λf(a)(f(x)). 89 Mengmeng Liu, Hongxing Liu Example 4.1. Let f : A1 → A2 be an Equasi-MV homomorphism. We can define θ = {(x,y) ∈ A1 ×A1 : f(x) = f(y)}, then θ is a congruence. Let A be an Equasi-MV algebra and θ be a congruence on A. We denote A/θ = {x/θ : x ∈ A}, where x/θ = {y ∈ A : 〈x,y〉 ∈ θ}. We define operations e, d, ⊕ on A/θ as follows: for any x,y ∈ A, x/θ e y/θ = (x e y)/θ, x/θ d y/θ = (x d y)/θ, x/θ ⊕y/θ = (x⊕y)/θ. Suppose x/θ 6 y/θ. Then (x e y)/θ > x/θ. For all z ∈ A, we have x/θ ⊕z/θ = (x⊕z)/θ 6 ((x e y)⊕z)/θ 6 (y ⊕z)/θ = y/θ ⊕z/θ. This proves that 〈A/θ,⊕,0/θ〉 is a commutative preordered semigroup and (x/θ⊕ y/θ)⊕0/θ = x/θ ⊕y/θ. For all x ∈ A, there exists a ∈ I(A) such that x 6 a. It is easily shown that a/θ is an idempotent element and x/θ 6 a/θ. Since A is an Equasi-MV algebra, we have that 〈[0,a],⊕,λa,0,a〉 is a quasi-MV algebra. And let θa = θ∩ ([0,a]× [0,a]) be an ideal congruence on 〈[0,a],⊕,λa,0,a〉. For any x/θa ∈ [0/θa,a/θa], we define λa/θa(x/θa) = λa(x)/θa. Then [0/θa,a/θa] is a quasi-MV algebra. Now we show that 〈[0/θ,a/θ],⊕,λa/θ,0/θ,a/θ〉 is a quasi-MV algebra. For all x/θ ∈ [0/θ,a/θ], there exists y/θ ∈ [0/θ,a/θ] such that x/θ ⊕y/θ = a/θ. It follows that 〈x ⊕ y,a〉 ∈ θ. And since x,y 6 a, we have 〈x ⊕ y,a〉 ∈ θa. That is, x/θa ⊕ y/θa = a/θa. Thus y/θa > λa(x)/θa and so y/θ > λa(x)/θ. This implies that λa/θ(x/θ) exists and equals to λa(x)/θ. It can be easily shown that 〈[0/θ,a/θ],⊕,λa/θ,0/θ,a/θ〉 is a quasi-MV algebra. Thus, 〈A/θ,⊕,0/θ〉 is an Equasi-MV algebra. And the map π : 〈A,⊕,0〉 −→ 〈A/θ,⊕,0/θ〉 defined by x 7−→ x/θ is an Equasi-MV homomorphism from A onto A/θ. Definition 4.3. Let A be an Equasi-MV algebra and I be a nonempty subset of A. We call I to be an ideal of A if the following conditions hold: (I1) 0 ∈ I; (I2) for all x,y ∈ I, then x⊕y ∈ I; (I3) x ∈ I and y 6 x imply y ∈ I. If I is an ideal of A and x ∈ A, we have x ∈ I iff x⊕0 ∈ I by (I3). Definition 4.4. Let A be an Equasi-MV algebra and I be a nonempty subset of A. If the following statements hold, I is a weak ideal of A: (W1) 0 ∈ I; (W2) for all x,y ∈ I, then x⊕y ∈ I; (W3) x ∈ I and y ∈ A imply x⊗y ∈ I. 90 On extended quasi-MV algebras Lemma 4.1. Let I be an ideal of an Equasi-MV algebra A. Then I is a weak ideal. Proof. Let I be an ideal of A and x ∈ I. If y ∈ A with y 6 x, there exists b ∈I(A) such that x,y 6 b. Then we have (x⊗y) e x = λb(λb(x)⊕λb(y)) e x = λb(λb(x)⊕λb(y)⊕λb(λb(x)⊕λb(y)⊕x)) = λb(λb(x)⊕λb(y)⊕λb(b)) = x⊗y. It follows that x⊗y 6 x. Thus x⊗y ∈ I and so I is a weak ideal of A.2 The converse of Lemma 4.1 is not true. For example, {0} is a weak ideal, but not an ideal. Proposition 4.1. Let I be a nonempty subset of an Equasi-MV algebra A and 0 ∈ I. Then I is an ideal iff for all x,y ∈ A, a ∈ I(A) with x,y 6 a, λa(x) ⊗y ∈ I and x ∈ I implies y ∈ I. Proof. =⇒: Let I be an ideal of A. For all x,y ∈ A and a ∈ I(A) with x,y 6 a, if λa(x)⊗y ∈ I and x ∈ I, we have (λa(x)⊗y)⊕x ∈ I. Since λa(y)⊕ ((λa(x)⊗y)⊕x) = λa(y)⊕ (λa(x⊕λa(y))⊕x) = λa(y)⊕ (λa(λa(x)⊕y)⊕y) (QMV4) = λa(y)⊕y ⊕λa(λa(x)⊕y) = a, we have y 6 (λa(x)⊗y)⊕x ∈ I and y ∈ I. ⇐=: For any x,y ∈ I and a ∈ I(A) with x 6 y and x,y 6 a, we have λa(x)⊗y = 0 ∈ I. Hence, y ∈ I is obtained from propositional conditions. And then λa(x)⊗ (x⊕y) = λa(x⊕λa(x⊕y)) = λa(x) e y 6 y ∈ I. Then λa(x)⊗ (x⊕y) ∈ I. It follows from x ∈ I that x⊕y ∈ I.2 Definition 4.5. Let A be an Equasi-MV algebra. We define a binary relation 4 as follows: for all x,y ∈ A, x 4 y iff x e y = x. The binary relation 4 satisfies antisymmetry and transitivity, but when x is a regular element, it satisfies reflexivity. Lemma 4.2. Let A be an Equasi-MV algebra and x,y ∈ A. Then x 4 y iff x 6 y and x ∈R(A). 91 Mengmeng Liu, Hongxing Liu Proof. If x 4 y, we have x e y = x and x e y = (x e y) ⊕ 0 = x ⊕ 0. It follows that x 6 y and x ⊕ 0 = x. Thus x ∈ R(A). Conversely, if x 6 y and x ∈R(A), we have x e y = x⊕0 = x and so x 4 y.2 Lemma 4.3. Let A be an Equasi-MV algebra and J ⊆ A. Then the following statements are equivalent: (i) J is a weak ideal of A; (ii) (1) if x,y ∈ J, then x⊕y ∈ J; (2) if x ∈ J, y 4 x, then y ∈ J. Proof. (i)=⇒(ii): Suppose x ∈ J and y 4 x. There exists b ∈I(A) such that x 6 b. Then x⊗ (λb(x)⊕y) = x e y ∈ J. Since y 4 x, we have x e y = y ∈ J. (ii)⇐=(i): For any x ∈ J, y ∈ A, there exists b ∈ I(A) such that x,y 6 b. Since x⊗y 6 x and x⊗y ∈R(A) by Lemma 4.2, we have x⊗y 4 x. Therefore, x⊗y ∈ J.2 Let A be an Equasi-MV algebra and H be a subset of A. The ideal generated by H is the smallest ideal of A containing H, denoted by 〈H〉. Lemma 4.4. Let A be an Equasi-MV algebra and H ⊆ A, then (i) 〈H〉={x∈A: there exist h1,· · ·,hn∈H,n ∈ Nsuch that x6h1⊕···⊕hn}; (ii) 〈0〉 is the smallest ideal of A; (iii) If I is an ideal of A and x ∈ A, we have 〈I ∪{x}〉 = {z ∈ A : z 6 a⊕n.x for some a ∈ I and n ∈ N}. Proof. (i) We write M ={x∈A : there existh1, · · ·,hn∈H,n∈N such that x6 h1⊕···⊕hn}. Then M is an ideal of A. Now we show that M is the smallest ideal of A containing H. Suppose M ′ is an ideal of A containing H. For any x ∈ M, there exist h1, · · · ,hn ∈ H such that x 6 h1 ⊕ ···⊕ hn. As H ⊆ M ′, we get x ∈ M ′ and so M ⊆ M ′. (ii) By (i) we obvious get the result.2 Definition 4.6. An ideal I of an Equasi-MV algebra A is maximal if for all x ∈ A\ I, 〈I ∪{x}〉 = A. Definition 4.7. Let A be an Equasi-MV algebra and θ be a congruence on A. θ is an ideal congruence if for all x,y ∈ A, (x⊕0)θ(y ⊕0) ⇒ xθy. Example 4.2. Let A be an Equasi-MV algebra and x,y ∈ A. A binary relation χ defined as follows: xχy iff x 6 y and y 6 x. It is easy to show that χ is compatible with ⊕. We now show that for all b ∈ I(A), χ∩([0,b]×[0,b]) is congruence on the quasi-MV algebra 〈[0,b],⊕,λb,0,b〉. Suppose 〈x,y〉 ∈ χ ∩ ([0,b] × [0,b]). It follows from 〈x,y〉 ∈ χ that x 6 y and y 6 x. Hence, λb(y) 6 λb(x) and λb(x) 6 λb(y). Therefore, 〈λb(x),λb(y)〉 ∈ χ ∩ ([0,b] × [0,b]). That is, χ is a congruence on A. As a result, χ is an ideal congruence. 92 On extended quasi-MV algebras Definition 4.8. Let A be an Equasi-MV algebra, I be an ideal of A and θ be an ideal congruence on A. We define two relations f(J) on A×A and g(θ) on A as follows: 〈x,y〉 ∈ f(J) iff there exists b ∈I(A) such that x⊗λb(y),y ⊗λb(x) ∈ J; g(θ) = 0/θ = {x ∈ A : xθ0}. Theorem 4.1. Let A be an Equasi-MV algebra, J be an ideal of A and θ be an ideal congruence on A. (i) f(J) is an ideal congruence on A; (ii) g(θ) is an ideal of A; (iii) J = g(f(J)); (iv) θ = f(g(θ)). Proof. (i) Obviously, f(J) is a congruence on A. Now we show that f(J) is an ideal congruence. Let 〈x⊕0,y ⊕0〉 ∈ f(J). There exists b ∈I(A) such that x,y 6 b. Then λb(x ⊕ 0) ⊗ (y ⊕ 0), λb(y ⊕ 0) ⊗ (x ⊕ 0) ∈ J. It follows that λb(x)⊗y=λb(x⊕0)⊗(y⊕0) ∈ J. Similarly, λb(y)⊗x ∈ J. Thus, 〈x,y〉 ∈ f(J). Therefore, f(J) is an ideal congruence on A. (ii) Suppose 〈x,0〉 ∈ θ and y 6 x. We have 〈λb(x),b〉 ∈ θ. That implies 〈λb(x)⊕y,b〉 ∈ θ and so 〈x⊗(λb(x)⊕y), x⊗b〉 ∈ θ. That is, 〈xey,x⊕0〉 ∈ θ. It follows from y 6 x that x e y = y ⊕0. Thus, 〈y ⊕0,x⊕0〉 ∈ θ. Since θ is an ideal congruence on A, we have 〈y,x〉 ∈ θ. This together with 〈0,x〉 ∈ θ implies that 〈y,0〉 ∈ θ and so y ∈ g(θ). Therefore, g(θ) is an ideal of A. (iii) It is easily seen that g(f(J)) = {x ∈ A : x⊕ 0 ∈ J}. For all x ∈ A, we have x ∈ J iff x⊕0 ∈ J. Thus g(f(J)) = {x ∈ A : x ∈ J}. (iv) For any x,y ∈ A, if 〈x,y〉 ∈ f(g(θ)), there exists b ∈ I(A) such that x,y 6 b, 〈λb(x)⊗y,0〉 ∈ θ and 〈λb(y)⊗x,0〉 ∈ θ. Then 〈(λb(x)⊗y)⊕x,0⊕x〉 ∈ θ. By (λb(x) ⊗ y) ⊕ x = x d y, we get 〈x d y,0 ⊕ x〉 ∈ θ. Similarly, we have 〈x d y,0 ⊕ y〉 ∈ θ. Thus, 〈0 ⊕x,0 ⊕ y〉 ∈ θ. Since θ is an ideal congruence on A, we have 〈x,y〉 ∈ θ. Therefore, f(g(θ)) ⊆ θ. Conversely, if 〈x,y〉 ∈ θ, there exists b ∈ I(A) such that x,y 6 b and so 〈y ⊗ λb(x),x ⊗ λb(x)〉 ∈ θ. This together with x ⊗ λb(x) = 0 implies 〈y ⊗ λb(x),0〉 ∈ θ. Similarly, 〈x⊗λb(y),0〉 ∈ θ. Thus, 〈x,y〉 ∈ f(g(θ)). Therefore, θ ⊆ f(g(θ)).2 Let I be an ideal of an Equasi-MV algebra A. The relation θI is defined as follows: for all x,y ∈ A, (x,y)∈θI ⇐⇒ ∃b ∈I(A) withx,y 6 bsuch that λb(λb(x)⊕y),λb(λb(y)⊕x) ∈ I. Proposition 4.2. Let A be an Equasi-MV algebra. If I is an ideal of A, the relation θI is an ideal congruence on A. Proof. Let I be an ideal of A. Suppose 〈x,y〉,〈y,z〉 ∈ θI. We have λb(λb(x)⊕ y), λb(λb(y)⊕x) ∈ I and λb(λb(z)⊕y), λb(λb(y)⊕z) ∈ I where b ∈I(A) such 93 Mengmeng Liu, Hongxing Liu that x,y,z 6 b. Since I is an ideal of A, we have λb(λb(x)⊕y)⊕λb(λb(y)⊕z) ∈ I and λb(λb(y) ⊕ x) ⊕ λb(λb(z) ⊕ y) ∈ I. And (λb(x) ⊕ z) ⊕ (λb(λb(x) ⊕ y) ⊕ λb(λb(y)⊕z)) = b. It follows that λb(λb(x)⊕z) ∈ I. Similarly, λb(λb(z)⊕x) ∈ I. Then 〈x,z〉 ∈ θI. The reflexivity and symmetry is clear. It is easy to prove that θI is compatible with ⊕. For all u ∈ I(A) such that x,y,z 6 u. Now, we show that θIu = θI ∩ ([0,u]× [0,u]) is a congruence on the quasi-MV algebra 〈[0,u],⊕,λu,0,u〉. Suppose 〈x,y〉 ∈ θIu , we have λu(λu(x)⊕ y), λu(λu(y)⊕x) ∈ I ∩ ([0,u]× [0,u]). Then (λu(x⊕z)⊕ (y ⊕z))⊕λu(λu(x)⊕y) =λu(x⊕z)⊕x⊕z ⊕λu(λu(y)⊕x) =λu(λu(x)⊕λu(z))⊕λu(z)⊕z ⊕λu(λu(y)⊕x) =u. It follows that λu(λu(x⊕z)⊕(y⊕z)) 6 λu(λu(x)⊕y) ∈ θI. Then λu(λu(x⊕z)⊕ (y⊕z)) ∈ θI. Similarly, λu(λu(y⊕z)⊕(x⊕z)) ∈ θI. Thus, 〈x⊕z,y⊕z〉 ∈ θIu . And 〈λu(x),λu(z)〉 ∈ θIu is obvious. Therefore, θI is a congruence on A. For each 〈x ⊕ 0,y ⊕ 0〉 ∈ θI, we have λb(λb(x ⊕ 0) ⊕ (y ⊕ 0)), λb(λb(y ⊕ 0) ⊕ (x⊕ 0)) ∈ I. That is, λb(λb(x) ⊕y), λb(λb(y) ⊕x) ∈ I. Thus 〈x,y〉 ∈ θI. Therefore, θI is an ideal congruence.2 Theorem 4.2. Let A be an Equasi-MV algebra. There is a one-to-one correspon- dence between the set of all ideals and the set of all ideal congruences. Proof. Let I be an ideal of A and θI be an ideal congruence induced by I. Now we show that I = 0/θI. Since 0 ∈ I, we have 〈x,0〉 ∈ θI, for all x ∈ I. It follows that x ∈ 0/θI. Conversely, suppose x ∈ 0/θI. There exists a ∈ I(A) such that x 6 a. By Proposition 4.1, since λa(x) ⊗ 0 ∈ I and 0 ∈ I, we have x ∈ I. Hence, I = 0/θI. Let θ be an ideal congruence on A. Let I = 0/θ. Suppose 〈x,y〉 ∈ θI. There exists a ∈ I(A) such that x,y 6 a and λb(λb(x) ⊕ y), λb(λb(y) ⊕ x) ∈ I = 0/θ. That is, 〈λb(λb(x) ⊕ y),0〉 ∈ θ and 〈λb(λb(y) ⊕ x),0〉 ∈ θ. Hence, 〈λb(λb(x) ⊕ y) ⊕ y,0 ⊕ y〉 ∈ θ and 〈λb(λb(y) ⊕ x) ⊕ x,0 ⊕ x〉 ∈ θ. Since λb(λb(x)⊕y)⊕y = λb(λb(y)⊕x)⊕x, we have 〈x⊕0,y⊕0〉 ∈ θ. And since θ is an ideal congruence on A, we have 〈x,y〉 ∈ θ. Conversely, let 〈x,y〉 ∈ θ. There exists a ∈ I(A) such that x,y 6 a. Then 〈λa(x),λa(y)〉 ∈ θ and 〈λa(x)⊗y,λa(y)⊗y〉 ∈ θ. Since λa(y)⊗y = 0, we have λa(x) ⊗ y ∈ 0/θ. Similarly, λa(y) ⊗ x ∈ 0/θ. That is, 〈x,y〉 ∈ θI. Therefore, θ = θI.2 Theorem 4.3. Let A be an Equasi-MV algebra. Then f(I)◦f(J) = f(J)◦f(I) is vaild, where I and J are ideals of A. 94 On extended quasi-MV algebras Proof. Suppose f(I),f(J) ∈ ConI(A) and 〈x,y〉 ∈ f(I)◦f(J) for x,y ∈ A. So there exists z ∈ A such that 〈x,z〉 ∈ f(I) and 〈z,y〉 ∈ f(J). There exists b ∈I(A) such that x,y,z 6 b. Let p be a ternary term defined as follows: pb(x,y,z) = (x⊗ (λb(y)⊕ (y e z))) d (z ⊗ (λb(y)⊕ (y e x))). Then (x⊗ (λb(z)⊕ (z e y))) d (y ⊗ (λb(z)⊕ (z e x))) f(I) pb(z,z,y) = y ⊕0 and (x⊗ (λb(z)⊕ (z e y))) d (y ⊗ (λb(z)⊕ (z e x))) f(J) pb(x,y,y) = x⊕0. Let (x⊗ (λb(z)⊕ (z e y))) d (y ⊗ (λb(z)⊕ (y e x))) = t, where t 6 b ∈I(A). It follows from 〈t,y⊕0〉 ∈ f(I) and 〈t,x⊕0〉 ∈ f(J) that (y ⊕0)⊗λb(t), λb(y ⊕0)⊗ t ∈ I; (x⊕0)⊗λb(t), λb(x⊕0)⊗ t ∈ J. Now, y⊗λb(t) 6 (y⊕0)⊗λb(t) ∈ I, x⊗λb(t) 6 (x⊕0)⊗λb(t) ∈ J. Similarly, λb(y)⊗t 6 λb(y⊕0)⊗t ∈ I, λb(x)⊗t 6 λb(x⊕0)⊗t ∈ J. Thus, 〈t,y〉 ∈ f(I) and 〈t,x〉 ∈ f(J). That is, 〈x,y〉 ∈ f(J)◦f(I).2 Lemma 4.5. If A is an Equasi-MV algebra, the lattice ConI(A) of ideal congru- ences on A is a sublattice of Con(A). Proof. Let I, J be two ideals of A. It is easy to prove that f(I ∩ J) = f(I)∩f(J). Now we show that f(I ∨J) = f(I)∨f(J). Since g(f(I ∨ J)) = I ∨ J and g(f(I)) ∨ g(f(J)) = I ∨ J, we claim that g(f(I) ∨ f(J)) = g(f(I)) ∨ g(f(J)). Let x ∈ g(f(I)) ∨ g(f(J)) such that x 6 y ⊕ z where y ∈ g(f(I)) and z ∈ g(f(J)). Then we get 〈y,0〉 ∈ f(I), 〈z,0〉 ∈ f(J) and 〈y,z〉 ∈ f(I)◦f(J) = f(I)∨f(J). It follows that 〈z⊕0,0〉 ∈ f(J), 〈y⊕z,z⊕0〉 ∈ f(I) and 〈y⊕z,0〉 ∈ f(I)◦f(J) = f(I)∨f(J). And then x 6 y ⊕z ∈ g(f(I)∨f(J)). Therefore, g(f(I))∨g(f(J) ⊆ g(f(I)∨f(J)). Conversely, for any x ∈ g(f(I) ∨ f(J)), we have 〈x,0〉 ∈ f(I) ∨ f(J) = f(I) ◦ f(J). Then there exist z ∈ A and b ∈ I(A) such that 〈x,z〉 ∈ f(I) and 〈z,0〉 ∈ f(J). And 〈x⊗λb(z),0〉 ∈ f(I), 〈z,0〉 ∈ f(J). Then x 6 (x⊗λb(z))⊕ z. Since x⊗λb(z) ∈ g(f(I)) and z ∈ g(f(J)), we have x ∈ g(f(I)) ∨g(f(J)). Thus, g(f(I)∨f(J)) ⊆ g(f(I))∨g(f(J)).2 Theorem 4.4. ConI(A) is distributive. Proof. By Theorem 4.2, we only need to prove that the lattice of ideals on A is distributive. Suppose I,J,K are ideals on A and x ∈ I ∩ (J ∨K). Then x ∈ I and x 6 y ⊕z, for some y ∈ J, z ∈ K. Hence, x 6 (x e y)⊕ (x e z). It follows from x e y ∈ I ∩J, x e z ∈ I ∩K that x ∈ (I ∩J)∨ (I ∩K).2 95 Mengmeng Liu, Hongxing Liu 5 Filters and prime ideals In this section, we introduce the notions of filters and prime ideals of Equasi- MV algebras. Moreover, we study some properties of them. We prove that every Equasi-MV algebra has at least one maximal ideal. Also, we get prime theorem on Equasi-MV algebras. Definition 5.1. Let 〈A,⊕,0〉 be an Equasi-MV algebra and F be a nonempty subset of A. F is called a filter if the following conditions are satisfied: (i) for all x,y ∈ A, if x 6 y and x ∈ F , then y ∈ F ; (ii) for all x,y ∈ F , then x⊗y ∈ F . Definition 5.2. We call a filter F is proper if F 6= A. A proper filter F is maximal, if for all x ∈ A\F , 〈F ∪{x}〉 = A. Let A be an Equasi-MV algebra. For x ∈ A and n ∈ N, we define x1 = x, · · · , xn = xn−1 ⊗x, n > 2. Proposition 5.1. Let A be an Equasi-MV algebra and F be a filter of A. Then IF is an ideal of A, where IF := {λa(x) : x ∈ F,∃a ∈I(A),x 6 a}. Proof. For all x ∈ A, we have x ∈ IF ⇐⇒ ∃a ∈I(A) s.t. x 6 a,λa(x) ∈ F. It is obvious that 0 ∈ IF . Suppose x,y ∈ IF . There exist a,b ∈ I(A) such that x 6 a and y 6 b. It follows λa(x), λb(y) ∈ F . Let c ∈ I(A) such that a,b 6 c. Then λc(x), λc(y) ∈ F by Proposition 3.1 (iv). That implies λc(x) ⊗λc(y) ∈ F . Since λc(x), λc(y) 6 c and λc(x)⊗λc(y) = λc(x⊕y), we have x⊕y ∈ IF . Suppose x,y ∈ A with x ∈ IF and y 6 x. There exists a ∈ I(A) such that x 6 a and λa(x) ∈ F . Since x,y ∈ [0,a] and y 6 x, we have λa(x) 6 λa(y). It implies λa(y) ∈ F and y ∈ IF . 2 In the following, we give an equivalent condition of maximal filters. Proposition 5.2. Let A be an Equasi-MV algebra and F be a proper filter of A. (i) For all x ∈ A, 〈F ∪{x}〉 = {z ∈ A : z > y ⊗xn,∃n ∈ N,y ∈ F}; (ii) F is a maximal filter iff for all x /∈ F , there exist n ∈ N and b ∈ I(A) with x 6 b such that λb(xn) ∈ F . Proof. (i) It is obvious. (ii) Let F be a maximal filter and x /∈ F . We have 0 ∈ 〈F ∪{x}〉 by (i) and so there exist n ∈ N and y ∈ F such that 0 = y⊗xn. There exists b ∈I(A) such that x,y 6 b. Then b = λb(y ⊗xn) = λb(y) ⊕λb(xn), it follows that y 6 λb(xn) and λb(xn) ∈ F . Conversely, for any x ∈ A \ F , there exist n ∈ N, b ∈ I(A) such that λb(xn) ∈ F . Then 0 = λb(xn) ⊗xn and 0 ∈ 〈F ∪{x}〉. It follows that 〈F ∪{x}〉 = A and F is a maximal filter.2 96 On extended quasi-MV algebras Lemma 5.1. Let F be a proper filter of an Equasi-MV algebra A. (i) If a ∈ F ∩I(A), we have a /∈ IF . (ii) If a ∈ F ∩I(A), then for all b ∈I(A) with a < b, we have λb(a) ∈ IF . (iii) If F is a maximal filter of A, then for all a ∈I(A), a /∈ IF implies a ∈ F . (iv) If J is a maximal ideal of A, then ∀a ∈I(A)\J =⇒ λb(a) ∈ J, where b ∈I(A) and a < b. (∗) (v) If J is an ideal of A satisfying (∗), then FJ is a filter of A, where FJ := {λa(x) : x ∈ J,a ∈I(A)\J,x < a}. Proof. (i) Suppose a ∈ F ∩I(A) and a ∈ IF . There exists b ∈ I(A) such that a 6 b and λb(a) ∈ F . It follows from λb(a), a ∈ F that 0 = a⊗λb(a) ∈ F , which is a contradiction. (ii) It is obvious. (iii) Let a ∈ I(A) and a /∈ IF . For all b ∈ I(A) with a 6 b, we have λb(a) /∈ F by Proposition 5.1. Suppose a /∈ F . Since F is a maximal filter, we have 〈F ∪{a}〉 = A. By Proposition 5.2, there exist n ∈ N and x ∈ F such that 0 = x⊗an. We have u ∈ I(A) such that x,a 6 u and 0 = x⊗an = x⊗u an. Since a ∈ I(A), we get an = a and so u = λu(x) ⊕ λu(a). It follows that x 6 λu(a) and λu(a) ∈ F , which is a contradiction. (iv) Suppose a ∈ I(A) and a /∈ J. For any b ∈ I(A) and a < b, we have λb(a) ∈ 〈J ∪{a}〉 = A. By Lemma 4.4, there exist n ∈ N and x ∈ J such that λb(a) 6 x⊕n.a. Since a,λb(a) ∈ [0,b], we have λb(a) = λb(a)⊕0 = λb(a) e (x⊕n.a) 6 (λb(a) e x)⊕ (λb(a) e n.a) (Lemma 2.4 (iii)) = λb(a) e x. It follows λb(a) 6 x ∈ J and so λb(a) ∈ J. (v) Suppose x,y ∈ A with x 6 y and x ∈ FJ. There exists a ∈I(A)\J such that x < a and λa(x) ∈ J. Let b ∈I(A) and a,y 6 b. We have λb(y) 6 λb(x) 6 λa(x) ⊕λb(a). By (iv), we have λb(a) ∈ J and λa(x) ⊕λb(a) ∈ J. That implies λb(y) ∈ J and y ∈ FJ. Let x,y ∈ FJ. There exist a,b ∈ I(A) \ J such that x 6 a, y 6 b and λa(x),λb(y) ∈ J. Let c ∈ I(A) and a,b 6 c. We have λc(a),λc(b) ∈ J by (iv) and λc(x) 6 λc(x)⊕0 = λa(x)⊕λc(a) ∈ J, λc(y) 6 λc(y)⊕0 = λb(y)⊕λc(b) ∈ J by Proposition 3.1. It follows that λc(x),λc(y) ∈ J and λc(x) ⊕ λc(y) ∈ J. Thus λc(λc(x)⊕λc(y)) ∈ FJ. That is, x⊗y = x⊗c y ∈ FJ.2 Definition 5.3. Let A be an Equasi-MV algebra and I be an ideal of A. We call I to be prime if for all x,y ∈ A, x e y ∈ I implies that x ∈ I or y ∈ I. Proposition 5.3. Let I be an ideal of an Equasi-MV algebra A. Then I is prime iff 97 Mengmeng Liu, Hongxing Liu for any x,y ∈ A, there exists a ∈I(A) with x,y 6 a such that λa(λa(x)⊕y) ∈ I or λa(λa(y)⊕x) ∈ I. Proof. ⇐=: Let π: A −→ A/I be the canonical projection and θ be an ideal congruence. If x e y ∈ I, we have (x e y)/θ = x/θ e y/θ ∈ π(I). Let x/θ = [i] or y/θ = [j], where i,j ∈ I. There exists a ∈ I(A) such that x,y,i,j 6 a, λa(x) ⊗ i ∈ I, λa(i) ⊗ x ∈ I or λa(y) ⊗ j ∈ I, λa(j) ⊗ y ∈ I. It follows from Proposition 4.1 that x ∈ I or y ∈ I. =⇒: For any x,y ∈ A, there exists a ∈I(A) such that x,y 6 a. We have (λa(x)⊕y)d(λa(y)⊕x) =λa(x)⊕y⊕λa(λa(x)⊕y⊕λa(λa(y)⊕x)) =λa(x)⊕λa(λa(x)⊕λa(λa(y)⊕x))⊕λa(λa(λa(x)⊕λa(λa(y)⊕x))⊕λa(y)) =λa(y)⊕x⊕λa(λa(y)⊕x⊕x)⊕λa(λa(λa(x)⊕λa(λa(y)⊕x))⊕λa(y)) =λa(x)⊕λa(λa(y)⊕x))⊕λa((λa(x)⊕λa(λa(y)⊕x))⊕y)⊕x⊕λa(λa(y)⊕x⊕x) =a. It follows λa((λa(x) ⊕ y) d (λa(y) ⊕ x)) = 0 ∈ I. That is, λa(λa(x) ⊕ y) e λa(λa(y)⊕x) = 0 ∈ I. Therefore, λa(λa(x)⊕y) ∈ I or λa(λa(y)⊕x) ∈ I.2 Example 5.1. Let A × M be an Equasi-MV algebra mentioned in Example 3.6. It can be easily proved that P = {0,b} is a prime ideal of a quasi-MV algebra A. Now we show that P ×M is a prime ideal of an Equasi-MV algebra A×M. Obviously, 〈0,0〉 ∈ P × M and 〈0,M〉 ⊕ 〈b,M〉 = 〈b,M〉 ∈ P × M. And for any 〈x,M〉 6 〈b,M〉, we have 〈x,M〉 ∈ A × M. Then P × M is an ideal of A × M. For any 〈x1,y1〉,〈x2,y2〉 ∈ A × M, suppose 〈x1,y1〉 e 〈x2,y2〉 = 〈x1 ex2,y1∧y2〉 ∈ P×M, we have x1 ∈ P or x2 ∈ P . That is, 〈x1,y1〉 ∈ P×M or 〈x2,y2〉 ∈ P ×M. Let A be a proper Equasi-MV algebra and a ∈I(A)\{0}. We define ↑ a = {x ∈ A : x > a}. Then ↑ a is a filter of A. Moreover, ↑ a is a proper filter of A. Proposition 5.4. Let F be a maximal filter of an Equasi-MV algebra A. Then IF = {λa(x) : x ∈ F,∃a ∈I(A),x 6 a} is a maximal ideal of A. Proof. We know that IF is an ideal of A by Proposition 5.1. As F 6= ∅, we have a ∈I(A)∩F and so a /∈ IF by Lemma 5.1 (i). Let J be an ideal of A and IF ⊆ J. Suppose a /∈ J and a ∈ I(A), we have a /∈ IF and so a ∈ F by Lemma 5.1 (iii). Then for any b ∈ I(A) with a 6 b, we have λb(a) ∈ IF ⊆ J. Hence, J satisfies condition (∗) in Lemma 5.1 (iv). It follows from Lemma 5.1 (iv) that FJ is a filter of A. 98 On extended quasi-MV algebras Suppose x ∈ F and w ∈I(A)\J. There exists u ∈I(A) such that x,w 6 u. Since J is a proper ideal, we have u /∈ J. It follows from the definition of IF that λu(x) ∈ IF ⊆ J and then x ∈ FJ. That implies F ⊆ FJ. Since F is a maximal filter, we have FJ = F or FJ = A. If FJ = A, then there exist x ∈ J and a ∈ I(A) such that x < a and λa(x) = 0, which is a contradiction. Thus FJ = F . By Lemma 5.1 (v), for all x ∈ J, there exists a ∈ I(A) \ J such that x < a and λa(x) ∈ FJ = F . Hence, we have x ∈ IF . That is, J ⊆ IF . Thus J = IF . This proves that IF is a maximal ideal of A.2 Theorem 5.1. Let A be a proper Equasi-MV algebra. Then A has at least one maximal ideal. Proof. Suppose 0 6= a ∈ A. Note that ↑ a is a filter and {0} 6=↑ a. By Zorn’s lemma, we know that the set of all filters that does not contain 0 has a maximal element, which is a maximal filter of A, denoted by F . It follows from Proposition 5.4 that IF is a maximal ideal.2 The following statement gives the prime theorem on Equasi-MV algebras. Theorem 5.2. Let I be a proper ideal of an Equasi-MV algebra A and a ∈ A\I. Then there exists a maximal ideal P which contains I and a ∈ A\P . Moreover, P is prime. Proof. Let M = {J : I ⊆ J,a /∈ J} where I,J are ideals of A. By Zorn’s lemma, M has a top element P . It follows from I ∈ M that M 6= ∅. We claim that P is prime. Suppose x e y ∈ P and x,y /∈ P . We have a ∈ 〈P ∪{x}〉 and a ∈ 〈P ∪{y}〉. Then there exist n ∈ N and u,v ∈ P such that a 6 u⊕n.x and a 6 v ⊕n.y. It follows that a 6 (u⊕n.x) e (v ⊕n.y) 6 (u⊕v ⊕n.x) e (u⊕v ⊕n.y). By Lemma 2.4 (iii), we have a 6 (u⊕v⊕n.x)e(u⊕v⊕n.y) = (u⊕v)⊕(n.xen.y) 6 (u⊕v)⊕n2.(xey) ∈ P. It follows that a ∈ P , which is a contradiction. Thus, we have x ∈ P or y ∈ P .2 6 Conclusion In this paper, we introduce the notion of Equasi-MV algebras, which are gen- eralizations of quasi-MV algebras. We study some basic properties of Equasi-MV algebras, such as ideals, ideal congruences and filters and investigate their mutual relationships. We show that there is a one-to-one correspondence between the set of all ideals and the set of all ideal congruences on an Equasi-MV algebra. And we also studied some results on maximal ideals and prime ideals. There are many topics that deserve further study. For example, (1) can any Equasi-MV algebra be embedded into an Equasi-MV algebra with a top element? 99 Mengmeng Liu, Hongxing Liu (2) Does any simple Equasi-MV algebra have a top element? (3) The author in- troduced ME-algebras and studied the categorical equivalence between equality algebras and abelian lattice-ordered groups in Liu [2019]. We will study the rela- tionships between monadic Equasi-MV algebras and monadic equality algebras. References F. Paoli A. Ledda, M. Konig and R. Giuntini. MV-algebras and quantum compu- tation. Studia Logica, 82(2):245–270, 2006. C. C. Chang. Algebraic analysis of many valued logics. Transactions of the American Mathematical society, 88(2):467–490, 1958. P. F. Conrad and M. R. Darnel. Generalized Boolean algebras in lattice-ordered groups. Order, 14(4):295–319, 1997. A. Dvurečenskij and O. Zahiri. On EMV-algebras. Fuzzy Sets and Systems, 373: 116–148, 2019. A. Ledda F. Bou, F. Paoli and H. Freytes. On some properties of quasi-MV alge- bras and √ ′ quasi-MV algebras. Part II. Soft Computing, 12(4):341–352, 2008. R. Giuntini F. Paoli, A. Ledda and H. Freytes. On some properties of quasi-MV algebras and √ ′ quasi-MV algebras. Reports Math. Log., 44:31–63, 2009. R. Giuntini G. Cattaneo, M. L. Dalla Chiara and R. Leporini. An unsharp logic from quantum computation. International Journal of Theoretical Physics, 43 (7):1803–1817, 2004. H. Liu. On categorical equivalences of equality algebras and monadic equality algebras. Logic Journal of the IGPL, 27(3):267–280, 2019. H. Liu. EBL-algebras. Soft Computing, 24(19):14333–14343, 2020. J. Liu and W. Chen. A non-commutative generalization of quasi-MV algebras. In 2016 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), pages 122–126. IEEE, 2016. R. Giuntini M. L. Dalla Chiara and R. Greechie. Reasoning in quantum theory: sharp and unsharp quantum logics, volume 22. Springer Science & Business Media, 2013. R. Giuntini M. L. Dalla Chiara and R. Leporini. Logics from quantum computa- tion. International Journal of Quantum Information, 3(02):293–337, 2005. 100