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Published: 10 March 2023

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1

Rajabisotudeh, F., and N. Kouhestani. "MV-pseudo metrics on MV-algebras." Annals of the University of Craiova, Mathematics and Computer Science Series 49, no. 1 (June 24, 2022): 35–51. http://dx.doi.org/10.52846/ami.v49i1.1443.

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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.

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2

DI NOLA, A., P. FLONDOR, and B. GERLA. "COMPOSITION ON MV-ALGEBRAS." Journal of Algebra and Its Applications 05, no. 04 (August 2006): 417–39. http://dx.doi.org/10.1142/s0219498806001818.

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In this paper we introduce an extension of MV-algebras obtained by adding a binary operation and a constant, with the aim of modelling composition of functions. The variety of Composition MV-algebra (CMV-algebra, for short) is defined and some results regarding ideals and congruences are stated. Further, we define modules over CMV-algebras showing that to any substitution corresponds an endomorphism of modules.

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3

Heubo-Kwegna, Olivier A., and Jean B. Nganou. "Radically principal MV-algebras." Mathematica Slovaca 73, no. 1 (February 1, 2023): 25–36. http://dx.doi.org/10.1515/ms-2023-0004.

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Abstract An MV-algebra A is radically principal if every prime ideal P of A is radically principal, i.e., there exists a principal ideal I of A such that Rad ( P ) = Rad ( I ) $ \text{Rad}(P)=\text{Rad}(I) $ . We investigate radically principal MV-algebras and provide some characterizations as well as some classes of examples. We prove a Cohen-like theorem, precisely, an MV-algebra is radically principal if and only if every maximal ideal is radically principal. It is also shown that the radically principal hyperarchemedian MV-algebras are the weakly finite ones and the radically principal Boolean algebras are the finite ones. Radically principal MV-algebras are also studied from the perspective of lattice-ordered groups.

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4

Wang, Jun Tao, Yan Hong She, and Ting Qian. "Study of MV-algebras via derivations." Analele Universitatii "Ovidius" Constanta - Seria Matematica 27, no. 3 (December 1, 2019): 259–78. http://dx.doi.org/10.2478/auom-2019-0044.

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AbstractThe 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.

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5

Meng, Biao Long, and Xiao Long Xin. "A Note of Filters in Effect Algebras." Chinese Journal of Mathematics 2013 (November 10, 2013): 1–4. http://dx.doi.org/10.1155/2013/570496.

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We investigate relations of the two classes of filters in effect algebras (resp., MV-algebras). We prove that a lattice filter in a lattice ordered effect algebra (resp., MV-algebra) does not need to be an effect algebra filter (resp., MV-filter). In general, in MV-algebras, every MV-filter is also a lattice filter. Every lattice filter in a lattice ordered effect algebra is an effect algebra filter if and only if is an orthomodular lattice. Every lattice filter in an MV-algebra is an MV-filter if and only if is a Boolean algebra.

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6

Alshehri, N. O. "Derivations of MV-Algebras." International Journal of Mathematics and Mathematical Sciences 2010 (2010): 1–7. http://dx.doi.org/10.1155/2010/312027.

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We introduce the notion of derivation for an MV-algebra and discuss some related properties. Using the notion of an isotone derivation, we give some characterizations of a derivation of an MV-algebra. Moreover, we define an additive derivation of an MV-algebra and investigate some of its properties. Also, we prove that an additive derivation of a linearly ordered MV-algebral is an isotone.

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7

Borzooei, R. A., Akefe Radfar, and Sogol Niazian. "Relationship Between Hyper MV -algebras and Hyperlattices." Annals of West University of Timisoara - Mathematics and Computer Science 54, no. 2 (December 1, 2016): 75–94. http://dx.doi.org/10.1515/awutm-2016-0016.

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Abstract Sh. Ghorbani, et al. [9], generalized the concept of MV -algebras and defined the notion of hyper MV -algebras. Now, in this paper, we try to prove that any hyper MV -algebra is a hyperlattice. First we prove that any hyper MV -algebra that satisfies the semi negation property is a hyperlattice. Then with a computer program, we show that any hyper MV -algebra of order less than 6, is a hyperlattice. Finally, we claim that this result is correct for any hyper MV -algebra.

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8

Dvurečenskij, Anatolij. "Pseudo MV-algebras are intervals in ℓ-groups." Journal of the Australian Mathematical Society 72, no. 3 (June 2002): 427–46. http://dx.doi.org/10.1017/s1446788700036806.

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AbstractWe show that any pseudo MV-algebra is isomorphic with an interval Γ(G, u), where G is an ℓ-group not necessarily Abelian with a strong unit u. In addition, we prove that the category of unital ℓ-groups is categorically equivalent with the category of pseudo MV-algebras. Since pseudo MV-algebras are a non-commutative generalization of MV-algebras, our assertions generalize a famous result of Mundici for a representation of MV-algebras by Abelian unital ℓ-groups. Our methods are completely different from those of Mundici. In addition, we show that any Archimedean pseudo MV-algebra is an MV-algebra.

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9

Chajda, Ivan, and Helmut Länger. "Residuation in non-associative MV-algebras." Mathematica Slovaca 68, no. 6 (December 19, 2018): 1313–20. http://dx.doi.org/10.1515/ms-2017-0181.

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Abstract It is well known that every MV-algebra can be converted into a residuated lattice satisfying divisibility and the double negation law. In a previous paper the first author and J. Kühr introduced the concept of an NMV-algebra which is a non-associative modification of an MV-algebra. The natural question arises if an NMV-algebra can be converted into a residuated structure, too. Contrary to MV-algebras, NMV-algebras are not based on lattices but only on directed posets and the binary operation need not be associative and hence we cannot expect to obtain a residuated lattice but only an essentially weaker structure called a conditionally residuated poset. Considering several additional natural conditions we show that every NMV-algebra can be converted in such a structure. Also conversely, every such structure can be organized into an NMV-algebra. Further, we study an a bit more stronger version of an algebra where the binary operation is even monotone. We show that such an algebra can be organized into a residuated poset and, conversely, every residuated poset can be converted in this structure.

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10

Belluce, L. P., and A. Di Nola. "Simplicial structures in MV-algebras and logic." Journal of Symbolic Logic 72, no. 2 (June 2007): 584–600. http://dx.doi.org/10.2178/jsl/1185803624.

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Classical logic, as is well known, can be analyzed in a great part by algebraic methods using the Lindenbaum algebra obtained from the formal system. For example the completeness theorem for this logic becomes equivalent to the semisimplicity of the obtained Lindenbaum algebra.Since Chang [4, 5], Łukasiewicz logic has also been analyzed algebraically through the associated Lindenbaum type algebra, that is the algebra of equivalence classes obtained from the relation of provable equivalence. In this case this algebra is an MV-algebra [4]. Once again logical notions have an algebraic counterpart, for example, completeness relates strongly to semisimplicity [4, 5]. However, unlike the classical case where the algebras in question are Boolean and always semisimple, not all MV-algebras are semisimple. This fact, in a sense, enriches the theory of MV-algebras.Now every MV-algebra can be considered a Lindenbaum type algebra, namely an algebra associated to Łukasiewicz logic with additional axioms. Thus we can carry over to any MV-algebra various logical notions such as (in) completeness, consistency, satisfiability, etc.Two important logical notions are those of “formal consequence” and “semantical consequence”. The former just says that a wff α is deducible from a set of wff via the axioms and rules of inference, while the latter just says that every evaluation that “satisfies” all the members of also “satisfies” α.Informally call these relations F, S respectively; consider them as binary relations, Fα and Sα.Now the completeness theorem just states F = S. Thus we can talk about an MV-algebra being “complete” provided the associated relations F, S are equal.

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11

BELLUCE, LAWRENCE P., ANTONIO DI NOLA, and GIACOMO LENZI. "ALGEBRAIC GEOMETRY FOR MV-ALGEBRAS." Journal of Symbolic Logic 79, no. 4 (December 2014): 1061–91. http://dx.doi.org/10.1017/jsl.2014.53.

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AbstractIn this paper we try to apply universal algebraic geometry to MV algebras, that is, we study “MV algebraic sets” given by zeros of MV polynomials, and their “coordinate MV algebras”. We also relate algebraic and geometric objects with theories and models taken in Łukasiewicz many valued logic with constants. In particular we focus on the structure of MV polynomials and MV polynomial functions on a given MV algebra.

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12

Dai, Songsong. "Quasi-MV algebras for complex fuzzy logic." AIMS Mathematics 7, no. 1 (2021): 1416–28. http://dx.doi.org/10.3934/math.2022083.

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<abstract><p>Complex fuzzy logic (CFL) is an emerging topic of research in fuzzy logic. Due to the circle structured codomain of the complex membership function, algebraic structure (including MV-algebra) in traditional fuzzy logic cannot be easily transplanted to CFL. Quasi-MV algebras are almost identical to MV algebras, except $ \alpha\oplus\textbf{0} = \alpha $ does not always hold. In this paper, our goal is to derive some algebraic structures for CFL. We first construct a quasi-MV algebra in complex fuzzy logic by introducing negation and truncated sum in the unit disc of the complex plane $ \textbf{S} $. Next we construct a $ \sqrt{\neg} $ quasi-MV algebra over $ \textbf{S} $ by adding an operation of square root of the negation. Moreover, implication connective and some derived connectives on $ \textbf{S} $ are introduced. Furthermore, we construct a quasi-Wajsberg algebra over $ \textbf{S} $ in which implication is a primitive connective. These algebraic structures are suitable for CFL.</p></abstract>

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13

Chakraborty, M. K., and Jayanta Sen. "MV-algebras embedded in a CL-algebra." International Journal of Approximate Reasoning 18, no. 3-4 (April 1998): 217–29. http://dx.doi.org/10.1016/s0888-613x(98)00007-3.

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14

RIEČAN, BELOSLAV. "ON THE SUM OF OBSERVABLES IN PRODUCT MV ALGEBRAS." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 07, no. 06 (December 1999): 671–77. http://dx.doi.org/10.1142/s0218488599000544.

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As an alternative to quantum logics MV algebras have been suggested2,20. In a quite general class of MV algebras there have been constructed sum of any two observables10,14,20. Recently the notion of product MV algebra has been introduced4,5,12,17. In any weakly σ-distributive product MV algebra the joint observable of any observables can be constructed and on the base also the sum of any observables. In this paper we show that the two concepts of constructing of the sum are equivalent.

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15

Forouzesh, F. "N-Fold Obstinate Ideals in MV-Algebras." New Mathematics and Natural Computation 12, no. 03 (October 5, 2016): 265–75. http://dx.doi.org/10.1142/s1793005716500186.

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In this paper, we introduce the notion of [Formula: see text]-fold obstinate ideals of an [Formula: see text]-algebra and investigate the relations between the obstinate ideals and the other ideals of an [Formula: see text]-algebras and also we consider the quotient algebras induced by [Formula: see text]-fold obstinate ideals and prove some related theorems. In addition, the extension theorem of [Formula: see text]-fold obstinate ideals are obtained. Also, several characterizations of these [Formula: see text]-fold ideals are given. Finally, we construct algorithm for studing the structure of the [Formula: see text]-fold obstinate ideals in finite [Formula: see text]-algebras.

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16

Xin, Xiao-Long, and Pu Wang. "States and Measures on Hyper BCK-Algebras." Journal of Applied Mathematics 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/397265.

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We define the notions of Bosbach states and inf-Bosbach states on a bounded hyper BCK-algebra(H,∘,0,e)and derive some basic properties of them. We construct a quotient hyper BCK-algebra via a regular congruence relation. We also define a∘-compatibledregular congruence relationθand aθ-compatibledinf-Bosbach stateson(H,∘,0,e). By inducing an inf-Bosbach states^on the quotient structureH/[0]θ, we show thatH/[0]θis a bounded commutative BCK-algebra which is categorically equivalent to an MV-algebra. In addition, we introduce the notions of hyper measures (states/measure morphisms/state morphisms) on hyper BCK-algebras, and present a relation between hyper state-morphisms and Bosbach states. Then we construct a quotient hyper BCK-algebraH/Ker(m)by a reflexive hyper BCK-idealKer(m). Further, we prove thatH/Ker(m)is a bounded commutative BCK-algebra.

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17

Dvurečenskij, Anatolij. "Central elements and Cantor-Bernstein's theorem for pseudo-effect algebras." Journal of the Australian Mathematical Society 74, no. 1 (February 2003): 121–44. http://dx.doi.org/10.1017/s1446788700003177.

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AbstractPseudo-effect algebras are partial algebras (E; +, 0, 1) with a partially defined addition + which is not necessary commutative and with two complements, left and right ones. We define central elements of a pseudo-effect algebra and the centre, which in the case of MV-algebras coincides with the set of Boolean elements and in the case of effect algebras with the Riesz decomposition property central elements are only characteristic elements. If E satisfies general comparability, then E is a pseudo MV-algebra. Finally, we apply central elements to obtain a variation of the Cantor-Bernstein theorem for pseudo-effect algebras.

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18

DI NOLA, A., A. DVUREČENSKIJ, and A. LETTIERI. "ON THE LOOMIS–SIKORSKI THEOREM FOR MV-ALGEBRAS WITH INTERNAL STATE." Journal of the Australian Mathematical Society 89, no. 3 (December 2010): 317–33. http://dx.doi.org/10.1017/s144678871100111x.

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AbstractIn Flaminio and Montagna [‘An algebraic approach to states on MV-algebras’, in: Fuzzy Logic 2, Proc. 5th EUSFLAT Conference, Ostrava, 11–14 September 2007 (ed. V. Novák) (Universitas Ostraviensis, Ostrava, 2007), Vol. II, pp. 201–206; ‘MV-algebras with internal states and probabilistic fuzzy logic’, Internat. J. Approx. Reason.50 (2009), 138–152], the authors introduced MV-algebras with an internal state, called state MV-algebras. (The letters MV stand for multi-valued.) In Di Nola and Dvurečenskij [‘State-morphism MV-algebras’, Ann. Pure Appl. Logic161 (2009), 161–173], a stronger version of state MV-algebras, called state-morphism MV-algebras, was defined. In this paper, we present the Loomis–Sikorski theorem for σ-complete MV-algebras with a σ-complete state-morphism-operator, showing that every such MV-algebra is aσ-homomorphic image of a tribe of functions with an internal state induced by a function where all the MV-operations are defined by points.

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19

Jakubík, Ján. "On complete $MV$-algebras." Czechoslovak Mathematical Journal 45, no. 3 (1995): 473–80. http://dx.doi.org/10.21136/cmj.1995.128535.

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20

Chajda, I., R. Halaš, J. Kühr, and A. Vanžurová. "Normalization of $MV$-algebras." Mathematica Bohemica 130, no. 3 (2005): 283–300. http://dx.doi.org/10.21136/mb.2005.134099.

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21

Dvurečenskij, Anatolij. "Loomis-sikorski theorem for σ-complete MV-algebras and ℓ-groups." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 68, no. 2 (April 2000): 261–77. http://dx.doi.org/10.1017/s1446788700001993.

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AbstractWe show that every σ-complete MV-algebra is an MV-σ-homomorphic image of some σ-complete MV- algebra of fuzzy sets, called a tribe, which is a system of fuzzy sets of a crisp set Ω containing 1Ω and closed under fuzzy complementation and formation of min {∑nfn, 1}. Since a tribe is a direct generalization of a σ-algebra of crisp subsets, the representation theorem is an analogue of the Loomis-Sikorski theorem for MV-algebras. In addition, this result will be extended also for Dedekind σ-complete ℓ-groups with strong unit.

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22

Borzooei, R. A., M. Shenavaei, A. Di Nola, and O. Zahiri. "On EMV-Semirings." Mathematica Slovaca 69, no. 4 (August 27, 2019): 739–52. http://dx.doi.org/10.1515/ms-2017-0265.

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Abstract The paper deals with an algebraic extension of MV-semirings based on the definition of generalized Boolean algebras. We propose a semiring-theoretic approach to EMV-algebras based on the connections between such algebras and idempotent semirings. We introduce a new algebraic structure, not necessarily with a top element, which is called an EMV-semiring and we get some examples and basic properties of EMV-semiring. We show that every EMV-semiring is an EMV-algebra and every EMV-semiring contains an MV-semiring and an MV-algebra. Then, we study EMV-semiring as a lattice and prove that any EMV-semiring is a distributive lattice. Moreover, we define an EMV-semiring homomorphism and show that the categories of EMV-semirings and the category of EMV-algebras are isomorphic. We also define the concepts of GI-simple and DLO-semiring and prove that every EMV-semiring is a GI-simple and a DLO-semiring. Finally, we propose a representation for EMV-semirings, which proves that any EMV-semiring is either an MV-semiring or can be embedded into an MV-semiring as a maximal ideal.

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23

Jakubík, Ján. "Sequential convergences on $MV$-algebras." Czechoslovak Mathematical Journal 45, no. 4 (1995): 709–26. http://dx.doi.org/10.21136/cmj.1995.128552.

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24

Harlenderová, Magdalena, and Jiří Rachůnek. "Modal operators on MV-algebras." Mathematica Bohemica 131, no. 1 (2006): 39–48. http://dx.doi.org/10.21136/mb.2006.134083.

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25

SHANG, YUN, XIAN LU, and RUQIAN LU. "Automata theory based on unsharp quantum logic." Mathematical Structures in Computer Science 19, no. 4 (August 2009): 737–56. http://dx.doi.org/10.1017/s0960129509007701.

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By studying two unsharp quantum structures, namely extended lattice ordered effect algebras and lattice ordered QMV algebras, we obtain some characteristic theorems of MV algebras. We go on to discuss automata theory based on these two unsharp quantum structures. In particular, we prove that an extended lattice ordered effect algebra (or a lattice ordered QMV algebra) is an MV algebra if and only if a certain kind of distributive law holds for the sum operation. We introduce the notions of (quantum) finite automata based on these two unsharp quantum structures, and discuss closure properties of languages and the subset construction of automata. We show that the universal validity of some important properties (such as sum, concatenation and subset constructions) depend heavily on the above distributive law. These generalise results about automata theory based on sharp quantum logic.

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26

Xin, Xiao, Xi Zou, and Jian Zhan. "Hyper BL-algebras." Filomat 32, no. 19 (2018): 6675–89. http://dx.doi.org/10.2298/fil1819675x.

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We put forth the concept of hyper BL-algebras which is a generalization of BL-algebras. We give some non-trivial examples and properties of hyper BL-algebras. Moreover, we introduce weak filters and weak deductive systems of hyper BL-algebras and study the relationships between them. Then we state and prove some theorems about weak filters and weak deductive systems. In particular, we define the concept of regular compatible congruence on hyper BL-algebras and construct the quotient structure in hyper BL-algebras. Finally, we discuss the conditions in which a quotient hyper BL-algebra is an MV-algebra.

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27

Ciungu, Lavinia Corina. "Monadic pseudo BE-algebras." Mathematica Slovaca 70, no. 5 (October 27, 2020): 1013–40. http://dx.doi.org/10.1515/ms-2017-0411.

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AbstractIn this paper we define the monadic pseudo BE-algebras and investigate their properties. We prove that the existential and universal quantifiers of a monadic pseudo BE-algebra form a residuated pair. Special properties are studied for the particular case of monadic bounded commutative pseudo BE-algebras. Monadic classes of pseudo BE-algebras are investigated and it is proved that the quantifiers on bounded commutative pseudo BE-algebras are also quantifiers on the corresponding pseudo MV-algebras. The monadic deductive systems and monadic congruences of monadic pseudo BE-algebras are defined and their properties are studied. It is proved that, in the case of a monadic distributive commutative pseudo BE-algebra there is a one-to-one correspondence between monadic congruences and monadic deductive systems, and the monadic quotient pseudo BE-algebra algebra is also defined.

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28

Dvurečenskij, Anatolij. "MV-Observables and MV-Algebras." Journal of Mathematical Analysis and Applications 259, no. 2 (July 2001): 413–28. http://dx.doi.org/10.1006/jmaa.2000.7409.

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29

Borzooei, Rajab, Narges Akhlaghinia, Xiao Xin, and Mona Kologani. "Constructing some logical algebras from EQ-algebras." Filomat 35, no. 8 (2021): 2747–60. http://dx.doi.org/10.2298/fil2108747b.

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EQ-algebras were introduced by Nov?ak in [16] as an algebraic structure of truth values for fuzzy type theory (FTT). Nov?k and De Baets in [18] introduced various kinds of EQ-algebras such as good, residuated, and lattice ordered EQ-algebras. In any logical algebraic structures, by using various kinds of filters, one can construct various kinds of other logical algebraic structures. With this inspirations, by means of fantastic filters of EQ-algebras we construct MV-algebras. Also, we study prelinear EQ-algebras and introduce a new kind of filter and named it prelinear filter. Then, we show that the quotient structure which is introduced by a prelinear filter is a distributive lattice-ordered EQ-algebras and under suitable conditions, is a De Morgan algebra, Stone algebra and Boolean algebra.

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30

Lewin, Renato, Marta Sagastume, and Pedro Massey. "MV*—Algebras." Logic Journal of IGPL 12, no. 6 (November 2004): 461–83. http://dx.doi.org/10.1093/jigpal/12.6.461.

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31

Riečanová, Zdenka, and Jiří Janda. "MAXIMAL SUBSETS OF PAIRWISE SUMMABLE ELEMENTS IN GENERALIZED EFFECT ALGEBRAS." Acta Polytechnica 53, no. 5 (October 24, 2013): 457–61. http://dx.doi.org/10.14311/ap.2013.53.0457.

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We show that in any generalized effect algebra (G;⊕, 0) a maximal pairwise summable subset is a sub-generalized effect algebra of (G;⊕, 0), called a summability block. If G is lattice ordered, then every summability block in G is a generalized MV-effect algebra. Moreover, if every element of G has an infinite isotropic index, then G is covered by its summability blocks, which are generalized MV-effect algebras in the case that G is lattice ordered. We also present the relations between summability blocks and compatibility blocks of G. Counterexamples, to obtain the required contradictions in some cases, are given.

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32

Gluschankof, Daniel. "Cyclic ordered groups and MV-algebras." Czechoslovak Mathematical Journal 43, no. 2 (1993): 249–63. http://dx.doi.org/10.21136/cmj.1993.128391.

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33

Jakubík, Ján. "Direct product decomposition of $MV$-algebras." Czechoslovak Mathematical Journal 44, no. 4 (1994): 725–39. http://dx.doi.org/10.21136/cmj.1994.128490.

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34

Hamidi, M., and R. A. Borzooei. "DERIVATIONS OF MV-ALGEBRAS FROM HYPER MV-ALGEBRAS." Honam Mathematical Journal 38, no. 3 (September 25, 2016): 643–59. http://dx.doi.org/10.5831/hmj.2016.38.3.643.

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35

Nganou, Jean B. "Stone MV-algebras and strongly complete MV-algebras." Algebra universalis 77, no. 2 (January 17, 2017): 147–61. http://dx.doi.org/10.1007/s00012-016-0421-0.

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36

Cignoli, Roberto, and Antoni Torrens Torrell. "Boolean Products of MV-Algebras: Hypernormal MV-Algebras." Journal of Mathematical Analysis and Applications 199, no. 3 (May 1996): 637–53. http://dx.doi.org/10.1006/jmaa.1996.0167.

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37

BUŞNEG, DUMITRU, DANA PICIU, and ANCA-MARIA DINA. "Ideals in residuated lattices." Carpathian Journal of Mathematics 37, no. 1 (February 5, 2021): 53–63. http://dx.doi.org/10.37193/cjm.2021.01.06.

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"The notion of ideal in residuated lattices is introduced in [Kengne, P. C., Koguep, B. B., Akume, D. and Lele, C., L-fuzzy ideals of residuated lattices, Discuss. Math. Gen. Algebra Appl., 39 (2019), No. 2, 181–201] and [Liu, Y., Qin, Y., Qin, X. and Xu, Y., Ideals and fuzzy ideals in residuated lattices, Int. J. Math. Learn & Cyber., 8 (2017), 239–253] as a natural generalization of that of ideal in MV algebras (see [Cignoli, R., D’Ottaviano, I. M. L. and Mundici, D., Algebraic Foundations of many-valued Reasoning, Trends in Logic-Studia Logica Library 7, Dordrecht: Kluwer Academic Publishers, 2000] and [Chang, C. C., Algebraic analysis of many-valued logic, Trans. Amer. Math. Soc., 88 (1958), 467–490]). If A is an MV algebra and I is an ideal on A then the binary relation x ∼I y iff x^{*}Ꙩ y; x Ꙩy^{*} ∈ I , for x; y ∈ A; is a congruence relation on A. In this paper we find classes of residuated lattices for which the relation ∼ I (defined for MV algebras) is a congruence relation and we give new characterizations for i-ideals and prime i-ideals in residuated lattices. As a generalization of the case of BL algebras (see [Lele, C. and Nganou, J. B., MV-algebras derived from ideals in BL-algebras, Fuzzy Sets and Systems, 218 (2013), 103–113]), we investigate the relationship between i-ideals and deductive systems in residuated lattices."

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38

Yang, Yongwei, and Kuanyun Zhu. "MV-algebras with pseudo MV-valuations." Journal of Intelligent & Fuzzy Systems 37, no. 2 (September 9, 2019): 2621–36. http://dx.doi.org/10.3233/jifs-182844.

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Nola, Antonio Di, and Anatolij Dvurečenskij. "MV-Test Spaces Versus MV-Algebras." Czechoslovak Mathematical Journal 54, no. 1 (March 2004): 189–203. http://dx.doi.org/10.1023/b:cmaj.0000027259.40606.48.

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Piciu, Dana. "Pseudo-MV algebra of fractions and maximal pseudo-MV algebra of quotients." Central European Journal of Mathematics 2, no. 2 (April 2004): 199–217. http://dx.doi.org/10.2478/bf02476540.

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CARAMELLO, OLIVIA, and ANNA CARLA RUSSO. "LATTICE-ORDERED ABELIAN GROUPS AND PERFECT MV-ALGEBRAS: A TOPOS-THEORETIC PERSPECTIVE." Bulletin of Symbolic Logic 22, no. 2 (June 2016): 170–214. http://dx.doi.org/10.1017/bsl.2015.47.

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AbstractWe establish, generalizing Di Nola and Lettieri’s categorical equivalence, a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MV-algebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classifying topos, three levels of bi-interpretability holding for particular classes of formulas: irreducible formulas, geometric sentences, and imaginaries. Lastly, by investigating the classifying topos of the theory of perfect MV-algebras, we obtain various results on its syntax and semantics also in relation to the cartesian theory of the variety generated by Chang’s MV-algebra, including a concrete representation for the finitely presentable models of the latter theory as finite products of finitely presentable perfect MV-algebras. Among the results established on the way, we mention a Morita-equivalence between the theory of lattice-ordered abelian groups and that of cancellative lattice-ordered abelian monoids with bottom element.

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Chajda, Ivan, and Helmut Länger. "Basic semirings." Mathematica Slovaca 69, no. 3 (June 26, 2019): 533–40. http://dx.doi.org/10.1515/ms-2017-0245.

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Abstract Basic algebras were introduced by Chajda, Halaš and Kühr as a common generalization of MV-algebras and orthomodular lattices, i.e. algebras used for formalization of non-classical logics, in particular the logic of quantum mechanics. These algebras were represented by means of lattices with section involutions. On the other hand, classical logic was formalized by means of Boolean algebras which can be converted into Boolean rings. A natural question arises if a similar representation exists also for basic algebras. Several attempts were already realized by the authors, see the references. Now we show that if a basic algebra is commutative then there exists a representation via certain semirings with involution similarly as it was done for MV-algebras by Belluce, Di Nola and Ferraioli. These so-called basic semirings, their ideals and congruences are studied in the paper.

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Wang, Jun Tao, Xiao Long Xin, and Arsham Borumand Saeid. "Very true operators on MTL-algebras." Open Mathematics 14, no. 1 (January 1, 2016): 955–69. http://dx.doi.org/10.1515/math-2016-0086.

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AbstractThe main goal of this paper is to investigate very true MTL-algebras and prove the completeness of the very true MTL-logic. In this paper, the concept of very true operators on MTL-algebras is introduced and some related properties are investigated. Also, conditions for an MTL-algebra to be an MV-algebra and a Gödel algebra are given via this operator. Moreover, very true filters on very true MTL-algebras are studied. In particular, subdirectly irreducible very true MTL-algebras are characterized and an analogous of representation theorem for very true MTL-algebras is proved. Then, the left and right stabilizers of very true MTL-algebras are introduced and some related properties are given. As applications of stabilizer of very true MTL-algebras, we produce a basis for a topology on very true MTL-algebras and show that the generated topology by this basis is Baire, connected, locally connected and separable. Finally, the corresponding logic very true MTL-logic is constructed and the soundness and completeness of this logic are proved based on very true MTL-algebras.

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Xin, Xiao Long, Yi Jun Li, and Yu Long Fu. "States on Pseudo-BCI Algebras." European Journal of Pure and Applied Mathematics 10, no. 3 (April 20, 2017): 455–72. http://dx.doi.org/10.29020/nybg.ejpam.v10i3.2779.

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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 $\wedge_1$, $\wedge_2$ in a local bounded pseudo-BCI algebra $A$ and two local operations $\vee_1$ and $\vee_2$ in $V(a)$ for $a\in M(A)$. We show that in a local $\wedge_1$($\wedge_2$)-commutative local bounded pseudo-BCI algebra $A$, $(V(A),\wedge_1,\vee_1)$($(V(A),\wedge_2,\vee_2)$) forms a lattice for all $a\in M(a)$. 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.

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Noje, D., and B. Bede. "Vectorial MV-algebras." Soft Computing - A Fusion of Foundations, Methodologies and Applications 7, no. 4 (February 1, 2003): 258–62. http://dx.doi.org/10.1007/s00500-002-0197-3.

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Di Nola, Antonio, Revaz Grigolia, and Ada Lettieri. "Projective MV-algebras." International Journal of Approximate Reasoning 47, no. 3 (March 2008): 323–32. http://dx.doi.org/10.1016/j.ijar.2007.05.009.

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Busaniche, Manuela, Leonardo Cabrer, and Daniele Mundici. "Polyhedral MV-algebras." Fuzzy Sets and Systems 292 (June 2016): 150–59. http://dx.doi.org/10.1016/j.fss.2014.06.015.

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Belluce, L. P., A. Di Nola, and G. Lenzi. "Hyperfinite MV-algebras." Journal of Pure and Applied Algebra 217, no. 7 (July 2013): 1208–23. http://dx.doi.org/10.1016/j.jpaa.2012.10.012.

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Ambrosio, Rosanna. "Strict MV-algebras." Journal of Mathematical Analysis and Applications 237, no. 1 (September 1999): 320–26. http://dx.doi.org/10.1006/jmaa.1999.6482.

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Galatos, Nikolaos, and Constantine Tsinakis. "Generalized MV-algebras." Journal of Algebra 283, no. 1 (January 2005): 254–91. http://dx.doi.org/10.1016/j.jalgebra.2004.07.002.

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