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In this paper, we apply the theory of modules on hoops and introduce two concepts of modules on hoops and provide special examples and interesting results. Both concepts are correct and logical. The first concept is very close to the definition of module in abstract algebra. In this case, we investigate some important results in modules such as sub-modules and quotient structures. But if we want to investigate the relationship between hoop-modules and other modules on logical algebraic structures such as B C K -modules and M V -modules, we need to define the second definition of hoop-modules. In this case, we can get that a B C K -modules and an M V -module from any hoop-module. [ABSTRACT FROM AUTHOR]

AbstractThe paper studies the interplay between MV-algebras, Bézout domains, and Abelian l -groups, which was initiated by Y. C. Yang in 2006. For each MV-algebra M=Γ(G,u) , we delve into some properties of a subset of a Bézout domain corresponding to M. We study Boolean elements and ideals within the MV-algebra, focusing on their correspondence to subsets of Bézout domains. This investigation aids us in classifying Bézout domains about directly indecomposable MV-algebras or subdirect products of MV-algebras. We establish that each MV-algebra associated with a Noetherian Bézout domain is finite. In conclusion, from a ring-theoretical perspective, we present characteristics of perfect MV-algebras, (H,1) -perfect MV-algebras, hyperarchimedean MV-algebras, and complete MV-algebras. [ABSTRACT FROM AUTHOR]

The theory of extensive categories determines in particular the notion of separable MV-algebra (equivalently, of separable unital lattice-ordered Abelian group). We establish the following structure theorem: An MV-algebra is separable if, and only if, it is a finite product of algebras of rational numbers—i.e., of subalgebras of the MV-algebra [ 0 , 1 ] ∩ Q. Beyond its intrinsic algebraic interest, this research is motivated by the long-term programme of developing the algebraic geometry of the opposite of the category of MV-algebras, in analogy with the classical case of commutative K -algebras over a field K. [ABSTRACT FROM AUTHOR]

The main goal of this paper is the study of the Boolean algebra of all characteristic elements in a unital ℓ -group and we investigate some topological properties of it in the case that the unital ℓ -group equipped with a link or positive filter topology. We also introduce the concept of Boolean region as a subset of a unital ℓ -group. [ABSTRACT FROM AUTHOR]

In this paper,  (⊙ , ∨) -multiderivations on an MV-algebra A are introduced, the relations between  (⊙ , ∨) -multiderivations and  (⊙ , ∨) -derivations are discussed. The set  MD (A)  of  (⊙ , ∨) -multiderivations on A can be equipped with a preorder, and  (MD (A) / ∼ , ≼)  can be made into a partially ordered set with respect to some equivalence relation ∼. In particular, for any finite MV-chain  L n ,  (MD (L n) / ∼ , ≼)  becomes a complete lattice. Finally, a counting principle is built to obtain the enumeration of  MD (L n) . [ABSTRACT FROM AUTHOR]

Let A be an MV-algebra. An (⊙ , ∨) -derivation on A is a map d : A → A satisfying: d (x ⊙ y) = (d (x) ⊙ y) ∨ (x ⊙ d (y)) for all x , y ∈ A . This paper initiates the study of (⊙ , ∨) -derivations on MV-algebras. Several families of (⊙ , ∨) -derivations on an MV-algebra are explicitly constructed to give realizations of the underlying lattice of an MV-algebra as lattices of (⊙ , ∨) -derivations. Furthermore, (⊙ , ∨) -derivations on a finite MV-chain are enumerated and the underlying lattice is described. [ABSTRACT FROM AUTHOR]

The convergence theorems play an important role in the theory of probability and statistics and in its application. In recent times, we studied three types of convergence of intuitionistic fuzzy observables, i.e., convergence in distribution, convergence in measure and almost everywhere convergence. In connection with this, some limit theorems, such as the central limit theorem, the weak law of large numbers, the Fisher–Tippet–Gnedenko theorem, the strong law of large numbers and its modification, have been proved. In 1997, B. Riečan studied an almost uniform convergence on D-posets, and he showed the connection between almost everywhere convergence in the Kolmogorov probability space and almost uniform convergence in D-posets. In 1999, M. Jurečková followed on from his research, and she proved the Egorov's theorem for observables in MV-algebra using results from D-posets. Later, in 2017, the authors R. Bartková, B. Riečan and A. Tirpáková studied an almost uniform convergence and the Egorov's theorem for fuzzy observables in the fuzzy quantum space. As the intuitionistic fuzzy sets introduced by K. T. Atanassov are an extension of the fuzzy sets introduced by L. Zadeh, it is interesting to study an almost uniform convergence on the family of the intuitionistic fuzzy sets. The aim of this contribution is to define an almost uniform convergence for intuitionistic fuzzy observables. We show the connection between the almost everywhere convergence and almost uniform convergence of a sequence of intuitionistic fuzzy observables, and we formulate a version of Egorov's theorem for the case of intuitionistic fuzzy observables. We use the embedding of the intuitionistic fuzzy space into the suitable MV-algebra introduced by B. Riečan. We formulate the connection between the almost uniform convergence of functions of several intuitionistic fuzzy observables and almost uniform convergence of random variables in the Kolmogorov probability space too. [ABSTRACT FROM AUTHOR]

We prove that for any finitely generated projective abelian ℓ -group G , whence for any finitely generated free abelian ℓ -group, the lattice of ℓ -ideals of G is a Yosida frame. This answers a problem posed by Martínez and Zenk (2006) in their paper [12]. [ABSTRACT FROM AUTHOR]

In this paper, with the aim of further investigating on B L-algebras, we introduced the concept of the annihilator of complement elements of a non-empty subset X of a B L-algebra, (⊥(X−)), and we proved that this set is a filter. Then we examined and studied ⊥(X−) completely and showed under what conditions it becomes the prime and maximal filter. Next, with the help of ⊥(X−), we obtained an equivalence condition for fantastic filters, and since fantastic filters are directly related to MValgebras, we were able to find an equivalence condition for checking MV-algebras. Also, we studied ⊥(X−) in various types of BL-algebras, including linearly ordered BL-algebras, Boolean algebra, Godel algebra, integral and special BL-algebras. In the last part of this article, with the help of the spectrum of fantastic filters, (FS(L)), we introduced a new BL-algebra and examined some of its features. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the annihilating-ideal graph of an MV-algebra (A, ⊕, ∗, 0). The algebraic structure of MV-algebras (especially Boolean algebras) are described by using the annihilating-ideal graph. The connections between the ideal theory of MV-algebras and graph theory are established, which promote the studying of the coloring of graphs. The annihilating-ideal graph AG(A) is a simple graph with the vertex set V (AG(A)) = {I ∈ I(A)\{h0i, A} | ∃J ∈ I ∗ (A) such that IJ = h0i} and the edge set E(AG(A)) = {I − J | IJ = h0i, where I, J ∈ V (AG(A)) and I 6= J}, where I(A) is the set of all ideals of A and I ∗ (A) = I(A)\{h0i}. We verify that AG(A) is connected with dmax(AG(A)) ≤ 3. And we characterize some MV-algebras with dmax(AG(A)) = 0 or 1, where dmax(AG(A)) is the diameter of AG(A). If | A |≤ 7, we show that AG(A) is either a null graph, or dmax(AG(A)) = 1. We restrict MV-algebras to Boolean algebras. The connections between AG(A) and Γ(A) are studied, where Γ(A) is the zero-divisor graph of A. We characterize the complete graph AG(A) and the star graph AG(A) by using ann(A\{1}) = {a ∈ A | a ⊙ b = 0 for all b ∈ A\{1}}, where ann(A\{1}) is the annihilator of A\{1}. Finally, we study the vertex coloring and girth of AG(A). We give two lower bounds and an upper bound for χ(AG(A)). [ABSTRACT FROM AUTHOR]

In this paper, (⊙,∨)-multiderivations on an MV-algebra A are introduced, the relations between (⊙,∨)-multiderivations and (⊙,∨)-derivations are discussed. The set MD(A) of (⊙,∨)-multiderivations on A can be equipped with a preorder, and (MD(A)/∼,≼) can be made into a partially ordered set with respect to some equivalence relation ∼. In particular, for any finite MV-chain Ln, (MD(Ln)/∼,≼) becomes a complete lattice. Finally, a counting principle is built to obtain the enumeration of MD(Ln).

Let S be an additively idempotent semiring and Mn(S) be the semiring of all n×n matrices over S. We characterize the conditions of when the semiring Mn(S) is congruence-simple provided that the semiring S is either commutative or finite. We also give a characterization of when the semiring Mn(S) is subdirectly irreducible for S being almost integral (i.e. xy + yx + x = x for all x,y ∈ S). In particular, we provide this characterization for the semirings S derived from the pseudo MV-algebras. [ABSTRACT FROM AUTHOR]

The set of zero divisors of an MV-algebra is investigated. It is proved that the set of all zero divisors of an MV-algebra A is the union of all prime ideals of A. Next, Z°-ideals and Z-ideals are introduced and their properties are studied. We examine the relationship between them and their relationship with the minimal prime ideals. We study their behavior under homomorphism of MV-algebras. Also, we investigate in what conditions the annihilator of an ideal is a Z°-ideal and Z-ideal. [ABSTRACT FROM AUTHOR]

It is introduced a new algebra (A , ⊗ , ⊕ , ∗ , ⇀ , 0 , 1) called L P G -algebra if (A , ⊗ , ⊕ , ∗ , 0 , 1) is L P -algebra (i.e. an algebra from the variety generated by perfect MV-algebras) and (A , ⇀ , 0 , 1) is a Gödel algebra (i.e. Heyting algebra satisfying the identity (x ⇀ y) ∨ (y ⇀ x) = 1) . The lattice of congruences of an L P G -algebra (A , ⊗ , ⊕ , ∗ , ⇀ , 0 , 1) is isomorphic to the lattice of Skolem filters (i.e. special type of MV-filters) of the MV-algebra (A , ⊗ , ⊕ , ∗ , 0 , 1) . The variety L P G of L P G -algebras is generated by the algebras (C , ⊗ , ⊕ , ∗ , ⇀ , 0 , 1) where (C , ⊗ , ⊕ , ∗ , 0 , 1) is Chang MV-algebra. Any L P G -algebra is bi-Heyting algebra. The set of theorems of the logic L P G is recursively enumerable. Moreover, we describe finitely generated free L P G -algebras. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce convex structures on MV-algebras such that the MV-operations are convexity preserving or weak convexity preserving. Therefore, we propose the concepts of paraconvex MV-algebras and weak convex MV-algebras. We give some characterizations of weak convex MV-algebras. Further, we show that the standard MV-algebra endowed with its interval convexity is a weak convex MV-algebra. In particular, a finite MV-chain endowed with a non-trivial convex structure is a weak convex MV-algebra iff the convex structure is precisely its interval convexity. Moreover, the direct product of finite weak convex MV-algebras is still a weak convex MV-algebra. Based on this, we further get that each finite MV-algebra endowed with its interval convexity is a weak convex MV-algebra. By using ideals, we introduce the ideal convexity on an MV-algebra which turns it to be a paraconvex MV-algebra. Finally, we discuss the separation axioms on weak convex MV-algebras. Communicated by Ángel del Río Mateos [ABSTRACT FROM AUTHOR]

In this paper, we review some of their related properties of derivations on MV-algebras and give some characterizations of additive derivations. Then we prove that the fixed point set of Boolean additive derivations and that of their adjoint derivations are isomorphic. In particular, we prove that every MV-algebra is isomorphic to the direct product of the fixed point set of Boolean additive derivations and that of their adjoint derivations. Finally we show that every Boolean algebra is isomorphic to the algebra of all Boolean additive (implicative) derivations. These results also give the negative answers to two open problems, which were proposed in [Fuzzy Sets and Systems, 303(2016), 97–113] and [Information Sciences, 178(2008), 307–316]. [ABSTRACT FROM AUTHOR]

The convergence theorems play an important role in the theory of probability and statistics and in its application. In recent times, we studied three types of convergence of intuitionistic fuzzy observables, i.e., convergence in distribution, convergence in measure and almost everywhere convergence. In connection with this, some limit theorems, such as the central limit theorem, the weak law of large numbers, the Fisher–Tippet–Gnedenko theorem, the strong law of large numbers and its modification, have been proved. In 1997, B. Riečan studied an almost uniform convergence on D-posets, and he showed the connection between almost everywhere convergence in the Kolmogorov probability space and almost uniform convergence in D-posets. In 1999, M. Jurečková followed on from his research, and she proved the Egorov’s theorem for observables in MV-algebra using results from D-posets. Later, in 2017, the authors R. Bartková, B. Riečan and A. Tirpáková studied an almost uniform convergence and the Egorov’s theorem for fuzzy observables in the fuzzy quantum space. As the intuitionistic fuzzy sets introduced by K. T. Atanassov are an extension of the fuzzy sets introduced by L. Zadeh, it is interesting to study an almost uniform convergence on the family of the intuitionistic fuzzy sets. The aim of this contribution is to define an almost uniform convergence for intuitionistic fuzzy observables. We show the connection between the almost everywhere convergence and almost uniform convergence of a sequence of intuitionistic fuzzy observables, and we formulate a version of Egorov’s theorem for the case of intuitionistic fuzzy observables. We use the embedding of the intuitionistic fuzzy space into the suitable MV-algebra introduced by B. Riečan. We formulate the connection between the almost uniform convergence of functions of several intuitionistic fuzzy observables and almost uniform convergence of random variables in the Kolmogorov probability space too.

BL-algebras are algebraic structures corresponding to Hajek's basic fuzzy logic. The aim of this paper is to analyze the structure of BL-algebras using commutative rings. Due to computational considerations, we are interested in the finite case. We present new ways to generate finite BL-algebras using commutative rings and provide summarizing statistics. Furthermore, we investigated BL-rings, i.e., commutative rings whose the lattice of ideals can be equipped with a structure of BL-algebra. A new characterization for these rings and their connections to other classes of rings is established. Furthermore, we give examples of finite BL-rings for which the lattice of ideals is not an MV-algebra and, using these rings, we construct BL-algebras with 2 r + 1 elements, r ≥ 2 , and BL-chains with k elements, k ≥ 4. In addition, we provide an explicit construction of isomorphism classes of BL-algebras of small n size ( 2 ≤ n ≤ 5 ). [ABSTRACT FROM AUTHOR]

In this paper using the connections between some subvarieties of residuated lattices, we investigated some properties of the lattice of ideals in commutative and unitary rings. We give new characterizations for commutative rings A in which Id(A) is an MV-algebra, a Heyting algebra or a Boolean algebra and we establish connections between these types of rings. We are very interested in the finite case and we present summarizing statistics. We show that the lattice of ideals in a finite commutative ring is a Boolean algebra or an MV-algebra (which is not Boolean). Using this result we generate the binary block codes associated to the lattice of ideals in finite commutative rings and we present a new way to generate all (up to an isomorphism) finite MV-algebras using rings. [ABSTRACT FROM AUTHOR]

In this paper, we provide a set-theoretic proof of the general representation theorem for MV-algebras, which was developed by Dubuc and Poveda in 2010. The theorem states that every MV-algebra is isomorphic to the MV-algebra of all global sections of its prime spectrum. We avoid using topos theory and instead rely on basic concepts from MV-algebras, topology and set theory. [ABSTRACT FROM AUTHOR]

In this study, we define the generalized symmetric bi-derivation and its trace on MV-algebras and give its examples. We investigate some properties of generalized symmetric bi-derivation's trace. We introduce isotone generalized symmetric bi-derivation and bi-additive generalized symmetric bi-derivation. We show that if A is a linearly ordered MV-algebra, Γ is a bi-additive generalized symmetric bi-derivation on A related to symmetric bi-derivation D and γ is a trace of Γ, then γ = 0 or γ1 = 1 [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of a (strong) hyper RL-ideal in hyper residuated lattices and give some properties and characterizations of them. Next, we characterize the (strong) hyper RL-ideals generated by a subset and give some characterizations of the lattice of these hyper RL-ideals. Particularly, we prove that this lattice is algebraic and compact elements are finitely generated hyper RL-ideals, and obtain some isomorphism theorems. Finally, we introduce the notion of nodal hyper RL-ideals in a hyper residuated lattice and investigate their properties. We prove that the set of nodal hyper RL-ideals is a complete Brouwerian lattice and under suitable operations is a Heyting algebra.

In this paper, we investigate the connection between ideals of semisimple MV-algebras and set-theoretic filters. In particular, we obtain that there is a bijection between closed ideals (with respect to a specified closure operation) and all filters on some associated set X. Finally, we establish that the space of closed prime ideals is homeomorphic to a well-described space having the Stone-Čech compactification βX as a subspace. [ABSTRACT FROM AUTHOR]

In this paper, we define the notions of MV-pseudo norm and MV-pseudo metric on MV-algebras and study some of their algebraic properties. The notion of uniform MV-algebra is also introduced and its relationship to MV-pseudo metrics is studied. [ABSTRACT FROM AUTHOR]

We initiate a study of two general concepts of semisimplicity for MV-algebras by replacing the standard MV-algebra [ 0 , 1 ] with an arbitrary MV-chain C . These generalised notions are called C -semisimple MV-algebras and m -semisimple MV-algebras. We obtain several of their characterisations and explore in more-depth the case of perfect MV-chains. [ABSTRACT FROM AUTHOR]

We introduce and investigate n -roots in the context of MV-algebras as a generalization of square roots introduced in [16]. We outline their main properties and establish that the class of MV-algebras with n -roots, MV n r , forms a variety. Next, we introduce the concept of strict n -roots and demonstrate an equivalence between MV n r and the class of n -divisible unital ℓ -groups. It helped us to show that each MV-algebra with an n -root is a direct product of an n -strict MV-algebra and a Boolean algebra. Finally, we delve into the connection between strongly atomless MV-algebras and MV-algebras with strict n -roots, demonstrating that in the context of MV n r , these concepts are equivalent. [ABSTRACT FROM AUTHOR]

In this article we consider C(X, [0, 1]) as an MV-algebra. We show that C(X, [0, 1]) is hyperarchimedean if and only if X is a finite union of spaces that only admit constant real-valued continuous functions. We also characterize the essential ideals and implicative ideals of C(X, [0, 1]). Lastly, we characterize the molecules of C(X, [0, 1]). [ABSTRACT FROM AUTHOR]

Topological indices are numerical values in graphs. Recently, they have been an attractive topic and multidisciplinary research area in mathematics, computer science, chemistry, pharmacy, and etc. In this work, we investigate the Forgotten index of zero divisor graphs of MValgebra. Firstly, using isomorphism between Ln and n-element MV-chain, we generalize the Forgotten topological index for zero divisor graph of Ln where n > 5. Then, the Forgotten index of zero divisor graph for MV-algebras which have cardinality 5,6,7,8 is computed with supporting samples. [ABSTRACT FROM AUTHOR]

EQ-algebras were introduced by Novak (2006) as an algebraic structure of truth values for fuzzy-type theory (FFT). Novák and De Baets (2009) introduced various kinds of EQ-algebras such as good, residuated, and IEQ-algebras. In this paper, we define the notion of (pre)ideal in bounded EQ-algebras (BEQ-algebras) and investigate some properties. Then, we introduce a congruence relation on good BEQ-algebras by using ideals, and then, we solve an open problem in Paad (2019). Moreover, we show that in IEQ-algebras, there is a one-to-one correspondence between congruence relations and the set of ideals. In the following, we characterize the generated preideal in BEQ-algebras, and by using this, we prove that the family of all preideals of a BEQ-algebra is a complete lattice. Then, we show that the family of all preideals of a prelinear IEQ-algebras is a distributive lattice and becomes a Heyting algebra. Finally, we show that we can construct an MV-algebra from the family of all preideals of a prelinear IEQ-algebra. [ABSTRACT FROM AUTHOR]

In this paper, first we define the notion of involutive operator on bounded involutive equality algebras and by using it, we introduce a new class of equality algebras that we called it a tense like equality algebra. Then we investigate some properties of tense like equality algebra. For two involutive bounded equality algebras and an equality homomorphism between them, we prove that the tense like equality algebra structure can be transfer by this equality homomorphism. Specially, by using a bounded involutive equality algebra and quotient structure of it, we construct a quotient tense like equality algebra. Finally, we investigate the relation between tense like equality algebras and tense MV-algebras.

In this paper, we define and characterize the notions of (implicative, maximal, prime) ideals in hoops. Then we investigate the relation between them and prove that every maximal implicative ideal of a $\vee$-hoop with double negation property is a prime one. Also, we define a congruence relation on hoops by ideals and study the quotient that is made by it. This notion helps us to show that an ideal is maximal if and only if the quotient hoop is a simple MV-algebra. Also, we investigate the relationship between ideals and filters by exploiting the set of complements.

The main goal of this paper is to give some representations of MV-algebras in terms of derivations. In this paper, we investigate some properties of implicative and difference derivations and give their characterizations in MV-algebras. Then, we show that every Boolean algebra (idempotent MV-algebra) is isomorphic to the algebra of all implicative derivations and obtain that a direct product representation of MV-algebra by implicative derivations. Moreover, we prove that regular implicative and difference derivations on MV-algebras are in one to one correspondence and show that the relationship between the regular derivation pair (d, g) and the Galois connection, where d and g are regular difference and implicative derivation on L, respectively. Finally, we obtain that regular difference derivations coincide with direct product decompositions of MV-algebras.

BL-algebras are algebraic structures corresponding to Hajek’s basic fuzzy logic. The aim of this paper is to analyze the structure of BL-algebras using commutative rings. Due to computational considerations, we are interested in the finite case. We present new ways to generate finite BL-algebras using commutative rings and provide summarizing statistics. Furthermore, we investigated BL-rings, i.e., commutative rings whose the lattice of ideals can be equipped with a structure of BL-algebra. A new characterization for these rings and their connections to other classes of rings is established. Furthermore, we give examples of finite BL-rings for which the lattice of ideals is not an MV-algebra and, using these rings, we construct BL-algebras with 2r+1 elements, r≥2, and BL-chains with k elements, k≥4. In addition, we provide an explicit construction of isomorphism classes of BL-algebras of small n size (2≤n≤5).

It is introduced an immune dynamic n-valued Łukasiewicz logic I D Ł n on the base of n-valued Łukasiewicz logic Ł n and corresponding to it immune dynamic M V n -algebra ( I D L n -algebra), 1 < n < ω , which are algebraic counterparts of the logic, that in turn represent two-sorted algebras (M , R , ◊) that combine the varieties of M V n -algebras M = (M , ⊕ , ⊙ , ∼ , 0 , 1) and regular algebras R = (R , ∪ , ; , ∗) into a single finitely axiomatized variety resembling R-module with "scalar" multiplication ◊ . Kripke semantics is developed for immune dynamic Łukasiewicz logic I D Ł n with application in immune system. [ABSTRACT FROM AUTHOR]

In recent years, A. Di Nola et al. studied the notions of MV-semiring and semimodules and investigated related results [9, 10, 12, 26]. Now in this paper, by using an MV-semiring and an MV-algebra, we introduce the new definition of MV-semimodule, study basic properties and find some examples. Then we study A-ideals on MV-semimodules and Q-ideals on MV-semirings, and by using them, we study the quotient structures of MV-semimodule. Finally, we present the notions of prime A-ideal, torsion free MV-semimodule and annihilator on MV-semimodule and we study the relations among them. [ABSTRACT FROM AUTHOR]

In order to investigate the derivation theory on MV-algebras further more, we give the notion of τ-additive derivations as a generalization of additive derivations in [29]. Then we discuss some related properties and characterizations of τ-additive derivations, and obtain the condition for a τ- additive derivation to be an endomorphism. Furthermore, some properties of τ-additive derivations related with the fixed set Δd(A) are given. [ABSTRACT FROM AUTHOR]

In this paper, we will explain the reason that filter topological characterization of 2-divisible abelian lattice ordered groups cannot be easily translated into MV-algebras. Then we prove that a 2-divisible MV-algebra A is semisimple if and only if it is Hausdorff with respect to the filter topology induced by the lattice filter of the form F = ⋃ n ∈ N A a , n for any a ∈ A which exceeds zero. We also show that an MV-algebra is simple if and only if it is Hausdorff with respect to any filter topology. Furthermore, an MV-algebra A is semisimple if and only if for any lattice filter F , which satisfies F ∩ I = ∅ for any proper MV-algebra ideal I , A is Hausdorff with respect to F. [ABSTRACT FROM AUTHOR]

In this paper we investigate filter topologies on MV-algebras. We give several equivalent conditions for a filter topology making an MV-algebra into a topological MV-algebra. Moreover, we determine all filter topologies on an MV-chain; in particular, it is shown that an MV-chain with a filter topology is always a topological MV-algebra. If an MV-algebra is compact and Hausdorff with respect to a filter topology, then it is a finite product of MV-chains, each of which is a finite or the standard MV-algebra. [ABSTRACT FROM AUTHOR]

In the paper, we introduce the concept of a locally complete EMV-algebra and a locally σ -complete EMV-algebra. They are generalizations of a complete and σ -complete MV-algebra, respectively. We will find some representations for locally complete EMV-algebras with atomic idempotent elements. We prove that each EMV-algebra of this type is a subdirect product of directly indecomposable complete EMV-algebras with top element. Also, we find a necessary and sufficient condition for a locally complete EMV-algebra when its representing EMV-algebra with the greatest element 1 is complete. Finally, we investigate the Cantor–Bernstein-type theorems for locally σ -complete EMV-algebras and σ -complete EMV-algebras. [ABSTRACT FROM AUTHOR]