George Georgescu

University of Bucharest | Unibuc · Department of Computer Science

In this paper we define and study the mz-elements of an algebraic quantale as an abstraction of the mz-ideals of a commutative ring, recently introduced by Ighedo and McGovern. By using a result of Banaschewski, we prove that the set zA of the mz-elements of a coherent quantale A is a coherent frame, as the image of a localic nucleus s : A → A. We...

The aim of this paper is to define an abstract quantale framework for extending some properties of the zip rings (studied by Faith, Zelmanowitz, etc.) and the weak zip rings (defined by Ouyang). By taking as prototype the quantale of ideals of a zip ring (resp. a weak zip ring) we introduce the notion of zipped quantale (resp. weakly zipped quantal...

The reticulation L(R) of a commutative ring R was introduced by Joyal in 1975, then the theory was developed by Simmons in a remarkable paper published in 1980. L(R) is a bounded distributive algebra whose main property is that the Zariski prime spectrum Spec(R) of R and the Stone prime spectrum SpecId (L(R)) of L(R) are homeomorphic. The construct...

The reticulation of a unital ring R is a bounded distributive lattice whose main property is that the Zariski prime spectrum of R is homeomorphic with the Stone prime spectrum of . In this paper we develop an axiomatic theory of reticulation in the abstract framework offered by the integral complete l-groupoids (= icl-groupoids). For an algebraic...

The commutator operation in a congruence-modular variety $\mathcal{V}$ allows us to define the prime congruences of any algebra $A\in \mathcal{V}$ and the prime spectrum $Spec(A)$ of $A$. The first systematic study of this spectrum can be found in a paper by Agliano, published in Universal Algebra (1993). The reticulation of an algebra $A\in \mathc...

An ideal I of a ring R is a lifting ideal if the idempotents of R can be lifted modulo I. A rich literature has been dedicated to lifting ideals. Recently, new algebraic and topological results on lifting ideals have been discovered. This paper aims to generalize some of these results to coherent quantales. We introduce the notion of lifting elemen...

This paper concerns some types of coherent quantale morphisms: Baer, minimalisant, quasi rigid, quasi r- and quasi r∗-quantale morphisms. Firstly, we study how the reticulation functor L( ⋅) preserves the properties that define these types of quantale morphisms. Secondly, we prove some characterization theorems for quasi rigid, quasi r- and qua...

The Lifting Idempotent Property ($LIP$) of ideals in commutative rings inspired the study of Boolean lifting properties in the context of other concrete algebraic structures ($MV$-algebras, commutative l-groups, $BL$-algebras, bounded distributive lattices, residuated lattices,etc.), as well as for some types of universal algebras (C. Muresan and t...

We know from a previous paper that the reticulation of a coherent quantale A is a bounded distributive lattice L(A) whose prime spectrum is homeomorphic to m-prime spectrum of A. This paper studies how the reticulation can be used for transferring some properties of bounded distributive lattices to quantales and vice versa. We shall illustrate this...

We investigate from an algebraic and topological point of view the minimal prime spectrum of a universal algebra, considering the prime congruences w.r.t. the term condition commutator. Then we use the topological structure of the minimal prime spectrum to study extensions of universal algebras that generalize certain types of ring extensions.

Several topologies can be defined on the prime, the maximal and the minimal prime spectra of a commutative ring; among them, we mention the Zariski topology, the patch topology and the flat topology. By using these topologies, Tarizadeh and Aghajani obtained recently new characterizations of various classes of rings: Gelfand rings, clean rings, abs...

In ring theory, the lifting idempotent property (LIP) is related to some important classes of rings: clean rings, exchange rings, local and semilocal rings, Gelfand rings, maximal rings, etc. Inspired by LIP, lifting properties were also defined for other algebraic structures: MV-algebras, BL-algebras, residuated lattices, abelian l-groups, congrue...

The lattices of fractions were introduced by Brezuleanu and Diaconescu in 1969. They used this concept in order to construct a Grothendieck-style duality for the category D 01 of bounded distributive lattices. Then the lattices of fractions are studied in connection with other themes in lattice theory: lattices schemas, localization of bounded dist...

We continue the study of the reticulation of a universal algebra initiated in \cite{retic}, characterizing morphisms which admit an image through the reticulation and investigating the kinds of varieties that admit reticulation functors.

In ring theory, the lifting idempotent property (LIP) is related to some important classes of rings: clean rings, exchange rings, local and semilocal rings, Gelfand rings,maximal rings, etc. Inspired by LIP, there were defined lifting properties for other algebraic structures: MV-algebras, BL- algebras, residuated lattices, abelian l-groups, congru...

In ring theory, the lifting idempotent property (LIP) is related to some important classes of rings: clean rings, exchange rings, local and semilocal rings, Gelfand rings,maximal rings, etc. Inspired by LIP, there were defined lifting properties for other algebraic structures: MV-algebras, BL-algebras, residuated lattices, abelian l-groups, congrue...

The reticulation of an algebra A is a bounded distributive lattice L(A) whose prime spectrum of filters or ideals is homeomorphic to the prime spectrum of congruences of A, endowed with the Stone topologies. We have obtained a construction for the reticulation of any algebra A from a semi–degenerate congruence–modular variety C in the case when the...

The reticulation of an algebra $A$ is a bounded distributive lattice ${\cal L}(A)$ whose prime spectrum of filters or ideals is homeomorphic to the prime spectrum of congruences of $A$, endowed with the Stone topologies. We have obtained a construction for the reticulation of any algebra $A$ from a semi-degenerate congruence-modular variety ${\cal...

In this paper, we extend properties Going Up and Lying Over from ring theory to the general setting of congruence-modular equational classes, using the notion of prime congruence defined through the commutator. We show how these two properties relate to each other, prove that they are preserved by finite direct products and quotients and provide al...

In previous work, we have introduced and studied a lifting property in congruence--distributive universal algebras which we have defined based on the Boolean congruences of such algebras, and which we have called the Congruence Boolean Lifting Property. In a similar way, a lifting property based on factor congruences can be defined in congruence--d...

We define lifting properties for universal algebras, which we study in this general context and then particularize to various such properties in certain classes of algebras. Next we focus on residuated lattices, in which we investigate lifting properties for Boolean and idempotent elements modulo arbitrary, as well as specific kinds of filters. We...

We introduce and study the Congruence Boolean Lifting Property (CBLP) for congruence--distributive universal algebras, as well as a property related to CBLP, which we have called $(\star )$. CBLP extends the so--called Boolean Lifting Properties (BLP) from MV--algebras, BL--algebras and residuated lattices, but differs from the BLP when particulari...

In this paper, we introduce the lifting properties for the Boolean elements of bounded distributive lattices with respect to the congru-ences, filters and ideals, we establish how they relate to each other and to significant algebraic properties, and we determine important classes of bounded distributive lattices which satisfy these lifting propert...

We define lifting properties for universal algebras, which we study in this general context and then particularize to various such properties in certain classes of algebras. Next we focus on residuated lattices, in which we investigate lifting properties for Boolean and idempotent elements modulo arbitrary, as well as specific kinds of filters. We...

In this paper we define the Boolean Lifting Property (BLP) for residuated lattices to be the property that all Boolean elements can be lifted modulo every filter, and study residuated lattices with BLP. Boolean algebras, chains, local and hyperarchimedean residuated lattices have BLP. BLP behaves interestingly in direct products and involutive resi...

Advanced computational technologies using many-valued logic algebras in highly complex systems. quantum logics and quantum automata are compared with the Boolean logic algebra of supercomputers and PCs.

This article deals with some probabilistic model theory for intuitionistic predicate logic. We introduce the notions of intuitionistic probability, probabilistic structure for intuitionistic predicate logic and model of an intuitionistic probability. We prove a Gaifman-style completeness theorem: any intuitionistic probability has a weak probabilis...

We study the forcing operators on MTL-algebras, an algebraic notion inspired by the Kripke semantics of the monoidal t -norm based logic (MTL). At logical level, they provide the notion of the forcing value of an MTL-formula. We characterize the forcing operators in terms of some MTL-algebras morphisms. From this result we derive the equality of th...

Bosbach states represent a way of probabilisticly evaluating the formulas from various (commutative or non-commutative) many-valued logics. They are defined on the algebras corresponding to these logics with values in $[0,1]$. Starting from the observation that in the definition of Bosbach states there intervenes the standard MV-algebra structure o...

111. Baianu, I. C., Georgescu, G., Glazebrook, J.F.., and Brown, R. 2010. Łukasiewicz-Moisil, Many--Valued Logic Algebras of Highly-Complex Systems. BRAIN-- Broad Research in Artificial Intelligence and Neuroscience, 1: 1-- 12.

This paper is a contribution to the algebraic logic of probabilistic models of Łukasiewicz predicate logic. We study the MV-states defined on polyadic MV-algebras and prove an algebraic many-valued version of Gaifman’s completeness theorem.

In this paper we define, inspired by ring theory, the class of maximal residuated lattices with lifting Boolean center and prove a structure theorem for them: any maximal residuated lattice with lifting Boolean center is isomorphic to a finite direct product of local residuated lattices.

Power structures are obtained by lifting some mathematical structure (operations, relations, etc.) from an universe X to its power set ${\mathcal{P}(X)}$ . A similar construction provides fuzzy power structures: operations and fuzzy relations on X are extended to operations and fuzzy relations on the set ${\mathcal{F}(X)}$ of fuzzy subsets of X...

In this paper we study the notion of forcing for Łukasiewicz predicate logic (Ł∀, for short), along the lines of Robinson’s forcing in classical model theory. We deal with both finite and infinite forcing. As regard to the former we prove a Generic Model Theorem for Ł∀, while for the latter, we study the generic and existentially complete standard...

We introduce tense LM_n-algebras and tense MV-algebras as algebraic structures for some tense many-valued logics. A representation theorem for tense LM_n-algebras is proved and the polynomial equivalence between tense LM_3-algebras (resp. tense LM_4-algebras) and tense MV_3-algebras (resp. tense MV$_4$-algebras) is established. We study the pairs o...

MV-algebras were introduced in 1958 by Chang [4] and they are models of Lukasiewicz infinite-valued logic. Chang gives a correspondence between the category of linearly ordered MV-algebras and the category of linearly ordered abelian l-groups. Mundici [10] extended this result showing a categorical equivalence between the category of the MV-algebra...

In this paper we define maximal MV-algebras, a concept similar to the maximal rings and maximal distributive lattices. We prove that any maximal MV-algebra is semilocal, then we characterize a maximal MV-algebra as finite direct product of local maximal MV-algebras.

We introduce the tense LMn-algebras and the tense MV - algebras as algebraic structures for some tense many-valued logics. A rep- resentation theorem for tense LMn-algebras is proved and the polynomially equivalence between tense LM3-algebras (resp. tense LM4-algebras) and tense MV3-algebras (resp. tense MV4-algebras) is established. We study the p...

In this paper we prove an extension theorem for submeasures defined on Łukasiewicz–Moisil algebras. This results generalize the Dobrakov extension theorem for submeasures on Boolean algebras.

MTL -algebras are algebraic structures for the Esteva-Godo monoidal t- norm based logic ( MTL ), a many-valued propositional calculus that formalizes the structure of the real interval (0,1), induced by a left-continuous t-norm. Given a com- plete MTL -algebra X, we define the weak forcing value |ϕ|X and the forcing value (ϕ)X, for any formula ϕ of...

We introduce the notion of n-nuanced MV-algebra by performing a Łukasiewicz–Moisil nuancing construction on top of MV-algebras. These structures extend both MV-algebras and Łukasiewicz–Moisil algebras, thus unifying two important types of structures in the algebra of logic. On a logical level, n-nuanced MV-algebras amalgamate two distinct approache...

A many-valued logic approach to transformations of neuronal networks during development and genetic networks in neoplastic tissue. Several new theorems are derived related to experimental observations on developing embryos and the initiation of neoplastic tissue.

Fundamental properties of N-valued logics are compared and eleven theorems are presented for their Logic Algebras, including Łukasiewicz–Moisil Logic Algebras represented in terms of categories and functors. For example, the Fundamental Logic Adjunction Theorem allows one to transfer certain universal, or global, properties of the Category of Boole...

A categorical, higher dimensional algebra and generalized topos framework for Łukasiewicz–Moisil Algebraic–Logic models of non-linear dynamics in complex functional genomes and cell interactomes is proposed. Łukasiewicz– Moisil Algebraic–Logic models of neural, genetic and neoplastic cell networks, as well as signaling pathways in cells are formula...

The paper develops a study of order convergence in Łukasiewicz-Moisil algebras. An axiomatical notion of distance (covering the pointwise and the Heyting distances) is provided, together with an associated notion of Cauchy sequence. Under natural hypotheses, the existence of Cauchy completions is proven. We analyze the connection to Boolean algebra...

This paper is concerned with the algebraic foundations of many-valued probability theory. We introduce and study a new notion of probability (state) on Łukasiewicz-Moisil algebras. This notion is parameterized by the considered logical implication (residuum) in Łukasiewicz-Moisil logic, knowing that there are several natural choices for the residuu...

We present in the paper a very concise but updated survey emphasizing the research done by Gr. C. Moisil and his school in algebraic logic.

We introduce and study a notion of logical convergence in residuated lattices (with operators). It is considered a convergence in similarity degree, rather than a bare order convergence – the lack of symmetry of residuated lattices brings our approach more related to the logical structure than to the set of truth values.

In this paper we study the pseudo-hoops, structures introduced by B. Bosbach in [6, 7] under the name of complementary semigroups. We prove some of their properties and we define the basic concepts of filter and normal filter. The lattice of normal filters is isomorphic with the lattice of congruences of a pseudo-hoop. We also study some important...

The lack of double negation and de Morgan properties makes fuzzy logic unsymmetrical. This is the reason why fuzzy versions of notions like closure operator or Galois connection deserve attention for both antiotone and isotone cases, these two cases not being dual. This paper offers them attention, comming to the following conclusions: – some kind...

Pseudo-BL algebras are non-commutative fuzzy structures which generalize BL-algebras and pseudo-MV algebras. In this paper we study the states on a pseudo-BL algebra. This concept is obtained by using the Bosbach condition for each of the two implications of a pseudo BL-algebra. We also propose a notion of conditional state for BL-algebras.

Weak pseudo BL-algebras (WPBL-algebras) are non-commutative fuzzy structures which arise from pseudo-t-norms (i.e. the non-commutative versions of triangular norms). In this paper, we study the pairs of weak negations on WPBL-algebras, extending the case of weak negations on Esteva–Godo MTL-algebras. A geometrical characterization of the pairs of w...

 Fuzzy Galois connections were introduced by Bělohlávek in [4]. The structure considered there for the set of truth values is a complete residuated lattice, which places the discussion in a “commutative fuzzy world”. What we are doing in this paper is dropping down the commutativity, getting the corresponding notion of Galois connection and general...

We introduce the notions of closure operator and closure system in a non-commutative fuzzy framework, where the structure of truth values is a generalized residuated lattice, L. We investigate the relationship between L-Galois connections and a weakened form of L-closure operators, which will eventually lead us to three ways of indicating a hierarc...

BL algebras were introduced by Hájek as algebraic structures for his Basic Logic. MV algebras, product algebras and Gödel algebras are particular cases of BL algebras. On the other hand, the pseudo-MV algebras extend the MV-algebras in the same way in which the arbitrary l-groups extend the abelian l-groups. Recently, there were introduced pseudo-B...

A classical (crisp) concept is given by its extent (a set of objects) and its intent (a set of properties). In commutative fuzzy logic, the generalization comes naturally, considering fuzzy sets of objects and properties. In both cases (the first being actually a particular case of the second), the situation is perfectly symmetrical: a concept is g...

Abstract. An MV-convergence is a convergence on an MV-algebra which renders the operations continuous. We show that such convergences on a given MV-algebra A are exactly the restrictions of the bounded -convergences on the abelian -group in which A appears as the unit interval. Thus the theory of -convergence and Cauchy structures transfers to MV-a...

Pseudo-BL algebras are noncommutative generalizations of BL-algebras and they include pseudo-MV algebras, a class of structures that are categorically equivalent to l-groups with strong unit. In this paper we characterize directly indecomposable pseudo-BL algebras and we define and study different classes of these structures: local, good, perfect,...

. In this paper we shall use the multipliers in order to define a sheaf on the prime spectrum of a Heyting algebra. This construction allows us to obtain a Grothendieck-style duality for Heyting algebras.

BL algebras were introduced by Hájek as algebraic structures for his Basic Logic. MV algebras, product algebras and Gödel algebras are particular cases of BL algebras. On the other hand, the pseudo-MV algebras extend the MV-algebras in the same way in which the arbitrary l-groups extend the abelian l-groups. Recently, there were introduced pseudo-B...

Since MV algebras are categorically equivalent to abelian l-groups with strong unit, we have started from arbitrary l-groups and thus we have obtained the more general notion of pseudo-MV algebra. We followed closely [1] and [9].

 BL algebras were introduced by Hájek as algebraic structures for his Basic Logic, starting from continuous t-norms on [0,1]. MV algebras, product algebras and Gödel algebras are particular cases of BL algebras. On the other hand, the pseudo-MV algebras extend the MV-algebras in the same way in which the arbitrary l-groups extend the abelian l-grou...

In this paper we define the polyadic Pavelka algebras as algebraic structures for Rational Pavelka predicate calculus (RPL∀). We prove two representation theorems which are the algebraic counterpart of the completness theorem for RPL∀.

We study some model theory for Gaifman probability structures. A classical result of Horn-Tarski concerning the extension of probabilities on Boolean algebras will allow us to prove some preservation theorems for probability structures, the model-companion of logical probability, etc., extending some classical results in eastern model theory.

We extend BCK algebras to pseudo-BCK algebras, as MV algebras and BL algebras were extended to pseudo-MV algebras and pseudo-BL algebras, respectively. We make the connection with pseudo-MV algebras and with pseudo-BL algebras.

BL algebras were introduced by P. Hájek as algebraic structures for his basic logic. MV algebras, product algebras and Gödel algebras are special cases of BL algebras. We present here some noncommutative extensions introduced in the last two years. Pseudo-MV algebras extend MV-algebras in the same way in which arbitrary l-groups extend Abelian l-gr...

The representation of algebras by Boolean products is a very general problem in universal algebra. In this paper we shall characterize the Boolean products of BL-chains, the weak Boolean products of local BL-algebras, and the weak Boolean products of perfect BL-algebras.

 In this paper a concept of probability defined on a Lukasiewicz-Moisil algebra is proposed. We take some steps in developing the theory, including an extension theorem and some results related to conditional probabilities on Lukasiewicz-Moisil algebras.

We define the monadic Pavelka algebras as algebraic structures induced by the action of quantifiers in rational Pavelka predicate logic. The main result is a representation theorem for these structures.

In this paper we shall prove that l-rings are categorally equivalent to the MV*-algebras, a subcategory of perfect MV-algebras. We shall use this equivalence in order to characterize l-rings as quotients of certain semirings of matrices over MV*-algebras. We shall establish a relation between l-ideals in l-rings and some ideals in MV*-algebras. Thi...

One of the basic principles of probability theory is that the set of the events of a trial is a Boolean algebra. It is the case when we consider that the trial follows the laws of classical logic. On the other hand, there exist many trials which are based on a many-valued logic. In this case one can accept the hypothesis that the set of the events...

We define a concept of probability on an n-valued Lukasiewicz-Moisilalgebra and we present some basic properties. The main result is an extension theorem for continuous probabilities, which is already known for probabilities defined on Boolean algebras and MV-algebas.

In [8, 11, 12] the class IRN was introduced in order to obtain the lattice-theoretic analogues of some results of Conrad (see e.g. [4]). The aim of these paper is to provide other useful constructions in the study of the structure of relatively normal lattices. The introduced notions and results are purely lattice-theoretic extensions of notions an...

The MV (many-valued) algebras were introduced by Chang (1958) as algebraic models of the infinite-valued Lukasiewicz logic. MV-algebras constitute a variety. For every n⩽2, the class MV_n of all n-valued algebras is a subvariety of the variety of all MV-algebras. Each variety MV_n is generated by the finite-chain MV-algebra...

An MV-space is a topological space X such that there exists an MV-algebra A whose prime spectrum Spec A is homeomorphic to X. The characterization of the MV-spaces is an important open problem. We shall prove that any projective limit of MV-spaces in the category of spectral spaces is an MV-space. In this way, we obtain new classes of MV-spaces rel...

The aim of this paper is to investigate the compact sheaf representation of MV-algebras in order to obtain a characterization of the dual of the category of MV-algebras. We also describe some classes of MV-algebras in terms of stalks of the Pierce sheaf of MV-algebras.

The compact (sheaf) representations of rings were studied by C. J. Mulvey. An abstract lattice-theoretic treatment of the compact representations was given by H. Simmons [J. Algebra 126, 493-531 (1989; Zbl 0708.18006)]. Thus we can define the compact representations for any variety of universal algebras. A well-known theorem of Gelfand and Kolmogor...

which occur in a much similar way in both cases of distributive lattices and of commutative rings. Namely, the existence of a certain spatial frame of ideals (the o-ideals for lattices [5], [7], the virginal ideals for rings [3]) which is in particular cases isomorphic to the frame of open sets of the maximal ideal space. This in turn, ensures inte...