Pseudo MV-algebras Research Articles

Some results on pseudo MV-algebras with square roots

The paper provides a study of pseudo MV-algebras with square roots. We introduce different notions of a square root on a pseudo MV-algebra, and present their main properties. We show that the class of pseudo-MV-algebras with square roots is a proper subvariety of the variety of pseudo MV-algebras. Then, we define a strict square root to classify the class of pseudo MV-algebras with square roots. We found a relationship between strongly atomless pseudo MV-algebras and strict pseudo MV-algebras. Finally, we investigate square roots on representable symmetric pseudo MV-algebras, and we present a complete characterization of a square root and a weak square root on a representable symmetric pseudo MV-algebra using addition in a unital ℓ-group. Some interesting examples are provided.

L-fuzzy algebraic substructure

This article aims to provide a method for defining L-fuzzy algebraic substructures on general algebras. Concretely, the properties of L-fuzzy sets are first reviewed, and their representations are then provided. Next, algebraic substructures are generalised as the closure systems on the power set of the algebra, and the properties of the prime and maximal elements in the above closure system are investigated. Based on these facts, L-fuzzy algebraic substructures with respect to the closure system are defined and studied. Two equivalence characterisations of the sup property of the ordered set L are provided using L-fuzzy substructures. Similarly, some properties of L-fuzzy prime and maximal substructures with respect to the closure system are discussed. Finally, to demonstrate the broad applicability of the theory of L-fuzzy algebraic substructures, the theory is applied to some specific algebraic structures, such as groups and pseudo MV-algebras.

Kites and representations of pseudo MV-algebras

We investigate the structure of perfect residuated lattices, focussing especially on perfect pseudo MV-algebras. We show that perfect pseudo MV-algebras can be represented as a generalised version of kites of Dvurečenskij and Kowalski, and that they are categorically equivalent to ℓ-groups with a distinguished automorphism. We then characterise varieties generated by kites and describe the lattice of these varieties as a complete sublattice of the lattice of perfectly generated varieties of perfect pseudo MV-algebras.

A short note on categorical equivalences of proper weak pseudo EMV-algebras

We study the class of weak pseudo EMV-algebras without top element that are a non-commutative generalization of MV-algebras, pseudo MV-algebras and generalized Boolean algebras. We present their categorical equivalences to a special category of pseudo MV-algebras with a fixed maximal and normal ideal as well as to a special category of unital l-groups with a fixed maximal and normal l-ideal.

Results in L-algebras

We study some classes of L-algebras and we give characterizations of commutative KL-algebras and CL-algebras. It is proved that any commutative KL-algebra is a BCK-algebra and any self-similar CL-algebra is commutative. The commutative ideals are defined and studied, and it is proved that a CL-algebra is commutative if and only if all its ideals are commutative. We also present the L-algebras arising from other structures, such as BCK-algebras, pseudo MV-algebras, pseudo BL-algebras, pseudo-hoops, bounded R $$\ell $$ -monoids, BE-algebras, Hilbert algebras.

Quantum B-algebras with involutions

The aim of this paper is to define and study the involutive and weakly involutive quantum B-algebras. We prove that any weakly involutive quantum B-algebra is a residuated poset. As an application, we introduce and investigate the notions of existential and universal quantifiers on involutive quantum B-algebras. It is proved that there is a one-to-one correspondence between the quantifiers on weakly involutive quantum B-algebras. One of the main results consists of proving that any pair of quantifiers is a monadic operator on weakly involutive quantum B-algebras. We investigate the relationship between quantifiers on bounded sup-commutative pseudo BCK-algebras and quantifiers on other related algebraic structures, such as pseudo MV-algebras and bounded Wajsberg hoops.

Monadic pseudo BE-algebras

Abstract In 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.

SEMISIMPLE, ARCHIMEDEAN, AND SEMILOCAL PSEUDO MV-ALGEBRAS

The concepts of semisimple, Archimedean, and semilocal pseudo MV-algebras are investigated and many interesting facts concerning them are given. Pseudo MV -algebras were introduced by G. Georgescu and A. Iorgulescu in (6) and in- dependently by J. Rachunek in (8) (there they are called generalized MV -algebras or, for short, GMV -algebras) as a non-commutative generalization of MV -algebras. This work was intended as an attempt to order some notions appearing in the theory of these al- gebras. Semisimple pseudo MV -algebras and Archimedean pseudo MV -algebras are ex- amples of such notions. In Section 3 we give some characterizations of semisimple pseudo MV -algebras. Archimedean pseudo MV -algebras are investigated and characterized in Sec- tion 4. It is shown that in the case of pseudo MV -algebras the notion of Archimedean is equivalent with the notion of Archimedean in the Belluce sense, that occurs in the theory of MV -algebras, and both are equivalent with the notion of semisimple. Section 5 is devoted to introduce and characterize semilocal pseudo MV -algebras, the concept generalizing a simi- lar one from the theory of MV -algebras. For the convenience of the reader, in Section 2 we give the relevant material needed in the sequel, thus making our exposition self-contained.

What are pseudo EMV-algebras?

In the paper, we present EMV-algebras as a common generalization of MV-algebras and generalized Boolean algebras where a top element is not assumed a priori. In addition, we present a non-commutative generalization of EMV-algebras, pseudo MV-algebras and of generalized Boolean algebras. We present main representation results showing a very close connection of pseudo EMV-algebra with pseudo MValgebras, and we give a categorical representation of the category opseudo EMV-algebras without top element. We study also states as analogs of finitely additive states, their topological properties, and we present an integral representation of states by σ-additive probability measures. The paper is a survey over papers [13]{[19].

On pseudo-CI algebras

In this paper, we introduce the notion of pseudo-CI algebras and investigate some of their properties. It is a generalization of the notion of pseudo-BE algebras, pseudo-BCK algebras and pseudo-MV algebras. We give and provide some conditions for a pseudo-CI algebra to be a pseudo-BE algebra. Also, we define the class of singular pseudo-CI algebras and show that any singular pseudo-CI algebra is a pseudo-BCI algebra. The adjoint group of a pseudo-CI algebra is defined and studied. Furthermore, distributive pseudo-CI algebras are discussed and showed that these coincide with distributive pseudo-BE algebras.

A semiring-like representation of lattice pseudoeffect algebras

In order to represent lattice pseudoeffect algebras, a non-commutative generalization of lattice effect algebras, in terms of a particular subclass of near semirings, we introduce in this article the notion of near pseudoeffect semiring. Taking advantage of this characterization, in the second part of the present work, we present, as an application, an alternative, rather straight as well as simple, explanation of the relationship between lattice pseudoeffect algebras and pseudo-MV algebras by means of a simplified axiomatization of generalized ?ukasiewicz semirings, a variety of non-commutative semirings equipped with two antitone unary operations.

States, state operators and quasi-pseudo-MV algebras

Quasi-pseudo-MV algebras (quasi-pMV algebras, for short) arising from quantum computational logics are the generalizations of both quasi-MV algebras and pseudo-MV algebras. In this paper, we introduce the notions of states, state-morphisms, state operators and state-morphism operators to quasi-pMV algebras. First, we present the related properties of states on quasi-pMV algebras and show that states and Bosbach states coincide on any quasi-pMV algebra. And then we investigate the relationship between state-morphisms and the normal and maximal ideals of quasi-pMV algebras. We prove state-morphisms and extremal states are equivalent. The existence of states on quasi-pMV algebras is also discussed. Finally, state operators and state-morphism operators are introduced to quasi-pMV algebras, and the corresponding structures are called state quasi-pMV algebras and state-morphism quasi-pMV algebras, respectively. We investigate the related properties of ideals under state operators and state-morphism operators. Meanwhile, we show that there is a bijective correspondence between normal $$\sigma $$ -ideals and ideal congruences on state quasi-pMV algebras.

Ideals and congruences in quasi-pseudo-MV algebras

Quasi-pseudo-MV algebras (quasi-pMV algebras, for short) were introduced both as the generalization of quasi-MV algebras and as the generalization of pseudo-MV algebras. In the present paper, we mainly investigate ideals and congruences in a quasi-pMV algebra. We present the properties of ideals of quasi-pMV algebras and investigate some special ideals. Furthermore, we study the normal ideals in detail and characterize the bijective correspondence between normal ideals and ideal congruences. Finally, we introduce weak ideals of a quasi-pMV algebra and show the relation existing between normal weak ideals and congruences.

Int-soft ideals of pseudo MV-algebras generated by a soft set

Int-soft ideals of pseudo MV-algebras generated by a soft set

When the lexicographic product of two po-groups has the Riesz decomposition property

We study conditions when a certain type of the Riesz Decomposition Property (RDP for short) holds in the lexicographic product of two po-groups. Defining two important properties of po-groups, we extend known situations showing that the lexicographic product satisfies RDP or even $${{\rm RDP}_1}$$ , a stronger type of RDP. We recall that a very strong type of RDP, $${{\rm RDP}_2}$$ , entails that the group is lattice ordered. RDP's of the lexicographic products are important for the study of lexicographic pseudo effect algebras, or perfect types of pseudo MV-algebras and pseudo effect algebras, where infinitesimal elements play an important role both for algebras as well as for the first order logic of valid but not provable formulas.

Orthocomplete pseudo MV-algebras

Pseudo MV-algebras are a non-commutative generalization of MV-algebras. The main purpose of the paper is to introduce and investigate orthocomplete pseudo MV-algebras. We use the concepts of projectable pseudo MV-algebras and large pseudo MV-subalgebras to introduce orthocomplete pseudo MV-algebras. Then we apply a generalization of the Mundici functor to an orthocompletion of an representable -group to prove that each representable pseudo MV-algebra has an orthocompletion. In particular, our results are valid also for MV-algebras.

A non-commutative generalization of Łukasiewicz rings

The goal of the present article is to extend the study of commutative rings whose ideals form an MV-algebra as carried out by Belluce and Di Nola [1] to non-commutative rings. We study and characterize all rings whose ideals form a pseudo MV-algebra, which shall be called here generalized Łukasiewicz rings. We obtain that these are (up to isomorphism) exactly the direct sums of unitary special primary rings.

Lexicographic pseudo MV-algebras

A lexicographic pseudo MV-algebra is an algebra that is isomorphic to an interval in the lexicographic product of a unital linearly ordered group with an arbitrary ℓ-group. We present conditions when a pseudo MV-algebra is lexicographic. We show that a key condition is the existence of a lexicographic ideal, or equivalently, a case when the algebra can be split into comparable slices indexed by elements of the interval [0,u] of some unital linearly ordered group (H,u). Finally, we show that fixing (H,u), the category of (H,u)-lexicographic pseudo MV-algebras is categorically equivalent to the category of ℓ-groups.

Pseudo MV-algebras and lexicographic product

We study algebraic conditions when a pseudo MV-algebra is an interval in the lexicographic product of an Abelian unital linearly ordered group and an ℓ-group that is not necessarily Abelian. We introduce two classes of pseudo MV-algebras which can be split into a system of comparable slices indexed by elements of an interval in an Abelian linearly ordered group. We show when such pseudo MV-algebras have a representation by a lexicographic product with an ℓ-group. Fixing a unital ℓ-group, we show that the category of such pseudo MV-algebras is categorically equivalent to the category of ℓ-groups.

ℍ-Perfect Pseudo MV-Algebras and Their Representations

Abstract We study ℍ-perfect pseudo MV-algebras, that is, algebras which can be split into a system of ordered slices indexed by the elements of an subgroup ℍ of the group of the real numbers. We show when they can be represented as a lexicographic product of ℍ with some ℓ-group. In addition, we show also a categorical equivalence of this category with the category of ℓ-groups.