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The first edition of this book appeared in 1981 as a direct continuation of Lectures of von Neumann Algebras (by Ş.V. Strătilă and L. Zsidó) and, until 2003, was the only comprehensive monograph on the subject. Addressing the students of mathematics and physics and researchers interested in operator algebras, noncommutative geometry and free probability, this revised edition covers the fundamentals and latest developments in the field of operator algebras. It discusses the group-measure space construction, Krieger factors, infinite tensor products of factors of type I (ITPFI factors) and construction of the type III_1 hyperfinite factor. It also studies the techniques necessary for continuous and discrete decomposition, duality theory for noncommutative groups, discrete decomposition of Connes, and Ocneanu's result on the actions of amenable groups. It contains a detailed consideration of groups of automorphisms and their spectral theory, and the theory of crossed products.

Gleason parts of a real function algebra boundaries for a real function algebra isometries of real function algebras symbols.

This paper is devoted to the description of complex finite-dimensional algebras of level two. We obtain the classification of algebras of level two in the variety of Leibniz algebras. It is shown that, up to isomorphism, there exist three Leibniz algebras of level two, one of which is solvable, and two of which are nilpotent. Moreover, we describe all algebras of level two in the variety of nilpotent algebras.

We present hyper effect algebras as a generalization of effect algebras. The result of the hyper summation of two mutually excluding events is not an element of the algebra but rather a subset (not necessarily a singleton) of the algebra. We present basic notions like states on hyper effect algebras. We present two standard examples of hyper effect algebras starting from effect algebras. We show how we can effectively generate finite models of hyper effect algebras and we point out problems with associativity. Finally, we provide a representation of any finite linearly ordered hyper effect algebra.

In the present paper we introduce the definition of a topological Segal algebra, which generalizes most of the earlier known definitions for Segal algebras. We also generalize some results about Segal algebras and algebras of continuous functions vanishing at infinity for the case of topological Segal algebras.

By using Artin-Wedderburn Theorem and the decomposition of central edepotent, several results about normality on closed subsets in standard table algebras are generalized to complex semi-simple algebras and the proofs are easier than the original ones.

In this paper, by using Gröbner-Shirshov bases theories, we prove that each countably generated associative differential algebra (resp., associative Ω-algebra, associative λ-differential algebra) can be embedded into a simple 2-generated associative differential algebra (resp., associative Ω-algebra, associative λ-differential algebra).

In this note we further explore the properties of universal enveloping algebras associated to a post-Lie algebra. Emphasizing the role of the Magnus expansion, we analyze the properties of group like-elements belonging to (suitable completions of) those Hopf algebras. Of particular interest is the case of post-Lie algebras defined in terms of solutions of modified classical Yang–Baxter equations. In this setting we will study factorization properties of the aforementioned group-like elements.

We correct a mistake in the description of orthoalgebras as pastings of Boolean algebras. We present a corrected structural theorem.

We classify three dimensional evolution algebras over a field having characteristic different from 2 and in which there are roots of orders 2, 3 and 7.

We compute the topological complexity for some important classes of noncommutative C*-algebras: AF algebras, AI algebras, and even Cuntz algebras.

The paper is a brief survey of the author’s results connected with the lattices of algebraic sets of universal algebras and with the operator of algebraic closure on the subsets of direct powers of basic sets of algebras.

A nonassociative algebra is defined to be zeropotent if the square of any element is zero. Zeropotent algebras are exactly the same as anticommutative algebras when the characteristic of the ground field is not two. The class of zeropotent algebras properly contains that of Lie algebras. In this paper, we give a complete classification of three-dimensional zeropotent algebras over an algebraically closed field of characteristic not equal to two. By restricting the result to the subclass of Lie algebras, we can obtain a classification of three-dimensional complex Lie algebras, which is in accordance with the conventional one.

In this work, we propose a new quantum structure--weak quantum MV algebras (wQMV algebras)--and define coupled bimonoids and strong coupled bimonoids. We find that the coupled bimonoids and strong coupled bimonoids are ring-like structures corresponding to lattice-ordered wQMV algebras and lattice-ordered QMV algebras, respectively. Using an automated reasoning tool, we give the smallest 4-element wQMV algebra but not a QMV algebra. We also show that lattice-ordered wQMV algebras are the real nondistributive generalization of MV algebras. Certainly, most important properties of quantum MV algebras (QMV algebras) are preserved by wQMV algebras. Furthermore, we can conclude that lattice-ordered wQMV algebras are the simplest unsharp quantum logical structures by far, based on which computation theory could be set up.

We investigate the support algebras of finite dimensional indecomposable modules over representation-infinite tame algebras of polynomial growth over an algebraically closed field.

In this paper, by considering the notion of equality algebra, which is introduced by Jenei in 18, as a possible algebraic semantic for fuzzy type theory, we study and show that there are relations among equality algebras and some of other logical algebras such as residuated lattice, MTL-algebra, BL-algebra, MV-algebra, Hertz-algebra, Heyting-algebra, Boolean-algebra, EQ-algebra and hoop-algebra. The aim of this paper is to find that under which conditions, equality algebras are equivalent to these logical algebras.

We establish that a set of sequences of real numbers is a lattice, with the partial order defined in a natural way. We obtain the least upper bound and greatest lower bound, present some algebraic ...

We investigate expansions of Heyting algebras (EHAs) in possession of a unary term describing the filters that correspond to congruences. Hasimoto proved that Heyting algebras equipped with finitely many (dual) normal operators have such a term, generalising a standard construction on finite-type boolean algebras with operators (BAOs). We utilise Hasimoto’s technique, extending the existence condition to a larger class of EHAs and some classes of double-Heyting algebras. Such a term allows us to characterise varieties with equationally definable principal congruences using a single equation. Moreover, in the presence of a dual pseudocomplement operation, discriminator varieties are characterised by a pair of equations. We also prove that a variety of dually pseudocomplemented EHAs with a normal filter term is semisimple if and only if it is a discriminator variety. This generalises two known results, one by Kowalski and Kracht for finite-type varieties of BAOs, and the other by the present author for dually pseudocomplemented Heyting algebras without additional operations.

Diagram algebras, in particular Brauer algebras, Birman–Murakami-Wenzl algebras and partition algebras, are used in representation theory and invariant theory of orthogonal and symplectic groups, in knot theory, in mathematical physics and elsewhere. Classifications are known when such algebras are semisimple, of finite global dimension or quasi-hereditary. We obtain a characterisation of the self-injective case, which is shown to coincide with the (previously also unknown) symmetric case. The main tool is to show that indecomposable self-injective algebras in general are derived simple, that is, their bounded derived module categories admit trivial recollements only. As a consequence, self-injective algebras are seen to satisfy a derived Jordan–Holder theorem.

We construct two kinds of infinite-dimensional 3-Lie algebras from a given commutative associative algebra, and show that they are all canonical Nambu 3-Lie algebras. We relate their inner derivation algebras to Witt algebras, and then study the regular representations of these 3-Lie algebras and the natural representations of the inner derivation algebras. In particular, for the second kind of 3-Lie algebras, we find that their regular representations are Harish-Chandra modules, and the inner derivation algebras give rise to intermediate series modules of the Witt algebras and contain the smallest full toroidal Lie algebras without center.

States on commutative basic algebras were considered in the literature as generalizations of states on MV-algebras. It was a natural question if states exist also on basic algebras which are not commutative. We answer this question in the positive and give several examples of such basic algebras and their states. We prove elementary properties of states on basic algebras. Moreover, we introduce the concept of a state-morphism and characterize it among states. For basic algebras which are the certain pastings of Boolean algebras the construction of a state-morphism is shown.

Generalizing the well known and exploited relation between Heyting and Nelson algebras to semi-Heyting algebras, we introduce the variety of semi-Nelson algebras. The main tool for its study is the construction given by Vakarelov. Using it, we characterize the lattice of congruences of a semi-Nelson algebra through some of its deductive systems, use this to find the subdirectly irreducible algebras, prove that the variety is arithmetical, has equationally definable principal congruences, has the congruence extension property and describe the semisimple subvarieties. Fil: Cornejo, Juan Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentina Fil: Viglizzo, Ignacio Dario. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentina

We study the measure semantics for propositional modal logics, in which formulas are interpreted in the Lebesgue measure algebra${\cal M}$, or algebra of Borel subsets of the real interval [0,1] modulo sets of measure zero. It was shown in Lando (2012) and Fernández-Duque (2010) that the propositional modal logic S4 is complete for the Lebesgue measure algebra. The main result of the present paper is that every logic L aboveS4 is complete for some subalgebra of ${\cal M}$. Indeed, there is a single model over a subalgebra of ${\cal M}$ in which all nontheorems of L are refuted. This work builds on recent work by Bezhanishvili, Gabelaia, & Lucero-Bryan (2015) on the topological semantics for logics above S4. In Bezhanishvili et al., (2015), it is shown that there are logics above that are not the logic of any subalgebra of the interior algebra over the real line, ${\cal B}$(ℝ), but that every logic above is the logic of some subalgebra of the interior algebra over the rationals, ${\cal B}$(ℚ), and the interior algebra over Cantor space, ${\cal B}\left( {\cal C} \right)$.

In this paper, the parakahler Hom-Lie algebras or phase space of Hom-Lie algebras in terms of Hom-left-symmetric algebras are studied. A structure theory of parakahler Hom-Lie algebras in terms of ...

The class of extended Lie-type algebras contains the ones of associative algebras, Lie algebras, Leibniz algebras, dual Leibniz algebras, pre-Lie algebras, and Lie-type algebras, etc. We focus on t...

Right-residuated binars and right-divisible residuated binars are defined as precursors of generalized hoops, followed by some results and open problems about these partially ordered algebras. Next we show that all complete homomorphic images of a complete residuated lattice A can be constructed easily on certain definable subsets of A. Applying these observations to the algebras of Hajek's basic logic (BL-algebras), we give an effective description of the HS-poset of finite subdirectly irreducible BL-algebras. The lattice of finitely generated BL-varieties can be obtained from this HS-poset by constructing the lattice of downward closed sets. These results are extended to bounded generalized BL-algebras using poset products and the duality between complete perfect Heyting algebras and partially ordered sets. We also prove that the number of finite generalized BL-algebras with n join-irreducible elements is, up to isomorphism, the same as the number of preorders on an n-element set, hence the same as the number of closure algebras (i.e., S4-modal algebras) with $$2^{n}$$2n elements. This result gives rise to a faithful functor from the category of finite GBL-algebras to the category of finite closure algebras that is full on objects, providing a novel connection between some substructural logics and classical modal logic. Finally, we show how generic satisfaction modulo theories solvers (SMT-solvers) can be used to obtain practical decision procedures for propositional basic logic and many of its extensions.

It is conjectured by Assem, Schiffler and Shramchenko in [3] that every cluster algebra is unistructural, that is to say, that the set of cluster variables determines uniquely the cluster algebra structure. In other words, there exists a unique decomposition of the set of cluster variables into clusters. This conjecture has been proven to hold true for algebras of Dynkin type or rank 2, see [3] . The aim of this paper is to prove it for algebras of type A ˜ . We use triangulations of annuli and algebraic independence of clusters to prove unistructurality for algebras arising from annuli, which are of type A ˜ . We also prove the automorphism conjecture from [3] for algebras of type A ˜ as a direct consequence.

We introduce a generalised notion of state as an additive map from a Boolean algebra of events to an arbitrary MV-algebra. Generalised states become unary operations in two-sorted algebraic structures that we call state algebras. Since these, as we show, form an equationally defined class of algebras, universal-algebraic techniques apply. We discuss free state algebras, their geometric representation, and their connection with the theory of affine representations of lattice groups.

Skew Boolean algebras for which pairs of elements have natural meets, called intersections, are studied from a universal algebraic perspective. Their lattice of varieties is described and shown to coincide with the lattice of quasi-varieties. Some connections of relevance to arbitrary skew Boolean algebras are also established.

In this paper, we transfer the notion of double extension, introduced by Medina and Revoy for quadratic Lie algebras [8], and extended by Benayadi and Baklouti for pseudo-euclidean Jordan algebras [1, 2], to the case of pseudo-euclidean alternative algebras. We show that every pseudo-euclidean alternative algebra, which is irreducible and neither simple nor nilpotent, is a suitable double extension. Moreover, we introduce the notion of generalized double extension of pseudo-euclidean alternative algebras by the one dimensional alternative algebra with zero product. This leads to an inductive classification of nilpotent pseudo-euclidean alternative algebras. A short review of the basics on alternative algebras and their connections to some other algebraic structures is also provided.

We study the algebra of differential operators on the triangular algebras and the upper triangular algebras. We further identify all the ideals of the algebra of differential operators on the upper triangular algebras.

In a new algebraic approach, conformal algebras and vertex algebras are extended to two-sorted structures, with an additional component encoding the logical properties of locality. Within these algebras, locality is expressed as an identity, without the need for existential quantifiers. Two-sorted conformal algebras form a variety of two-sorted algebras, an equationally-defined class, and free conformal algebras are given by standard universal algebraic constructions. The variety of two-sorted conformal algebras is equivalent to a Mal’tsev variety of single-sorted algebras. Motivated by a question of Griess, subalgebras of reducts of conformal algebras are shown to satisfy a set of quasi-identities. The class of two-sorted vertex algebras does not form a variety, so open problems concerning the nature of that class are posed.

Motivated by the definition of semi-Nelson algebras, a propositional calculus called semi-intuitionistic logic with strong negation is introduced and proved to be complete with respect to that class of algebras. An axiomatic extension is proved to have as algebraic semantics the class of Nelson algebras. Fil: Cornejo, Juan Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentina Fil: Viglizzo, Ignacio Dario. Universidad Nacional del Sur. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina

Let $P_n=k[x_1,x_2,\ldots,x_n]$ be the polynomial algebra over a field $k$ of characteristic zero in the variables $x_1,x_2,\ldots,x_n$ and $\mathscr{L}_n$ be the left-symmetric Witt algebra of all derivations of $P_n$. We describe all right operator identities of $\mathscr{L}_n$ and prove that the set of all algebras $\mathscr{L}_n$, where $n\geq 1$, generates the variety of all left-symmetric algebras. We also describe a class of general (not only right operator) identities for $\mathscr{L}_n$., 14 pages

In this paper we investigate the class of MV-algebras equipped with two quantifiers which commute as a natural generalization of diagonal-free two-dimensional cylindric algebras (see Henkin et al., in Cylindric algebras, 1985). In the 40s, Tarski first introduced cylindric algebras in order to provide an algebraic apparatus for the study of classical predicate calculus. The diagonal–free two-dimensional cylindric algebras are special cylindric algebras. The treatment here of MV-algebras is done in terms of implication and negation. This allows us to simplify some results due to Di Nola and Grigolia (Ann Pure Appl Logic 128(1-3):125–139, 2004) related to the characterization of a quantifier in terms of some special sub-algebra associated to it. On the other hand, we present a topological duality for this class of algebras and we apply it to characterize the congruences of one algebra via certain closed sets. Finally, we study the subvariety of this class generated by a chain of length n + 1 (n < ω). We prove that the subvariety is semisimple and we characterize their simple algebras. Using a special functional algebra, we determine all the simple finite algebras of this subvariety.

Extension of algebras can be constructed through various manners. In this paper, we introduce a new way of construction to extend a given algebra of dimension , which is isomorphic to the real Clifford algebra , to another algebra of dimension over the real numbers. And then, we generalize the construction over a -grading algebra. Moreover, we will show the relation between the new constructed algebras and Clifford algebras.

We provide a characterization for the endomorphism algebra of a generator to be a Morita algebra. As an application, n-Auslander algebras which are Morita algebras simultaneously are characterized.

The recent method of integration of non-stationary dynamical systems admitting nonlinear superpositions is applied to the three-dimensional dynamical systems associated with three-dimensional Vessiot-Guldberg-Lie algebras L 3 . The investigation is based on Bianchi’s classification of real three-dimensional Lie algebras and realizations of these algebras in the three-dimensional space. Enumeration of the Vessiot-Guldberg-Lie algebras L 3 allows to classify three-dimensional dynamical systems admitting nonlinear superpositions into thirty one standard types by introducing canonical variables. Twenty four of them are associated with solvable Vessiot-Guldberg-Lie algebras and can be reduced to systems of first-order linear equations. The remaining seven standard types are nonlinear. Integration of the latter types is an open problem.

In this paper, skew-morphisms, which are extensively studied in graph theory, are considered in the setting of matrix algebras. Different properties of skew-morphisms are obtained and their classification in some specific cases is given.

We show that the usual axiom system of quasi polyadic equality algebras is strongly redundant. Then, so called non-commutative quasi-polyadic equality algebras are introduced (), in which, among others, the commutativity of cylindrifications is dropped. As is known, quasi-polyadic equality algebras are not representable in the classical sense, but we prove that algebras in are representable by quasi-polyadic relativized set algebras, or more exactly by algebras in .

The method of direct computation of the universal (fibred) product in the category of commutative associative algebras of finite type with unity over a field is given and proven. The field of coefficients is not supposed to be algebraically closed and can be of any characteristic. Formation of fibred product of commutative associative algebras is an algebraic counterpart of gluing algebraic schemes by means of some equivalence relation in algebraic geometry. If initial algebras are finite-dimensional vector spaces, the dimension of their product obeys a Grassmann-like formula. A finite-dimensional case means geometrically the strict version of adding two collections of points containing a common part. The method involves description of algebras by generators and relations on input and returns similar description of the product algebra. It is "ready-to-eat"\, even for computer realization. The product algebra is well-defined: taking other descriptions of the same algebras leads to isomorphic product algebra. Also it is proven that the product algebra enjoys universal property, i.e. it is indeed a fibred product. The input data are a triple of algebras and a pair of homomorphisms \(A_1\stackrel{f_1}{\to}A_0\stackrel{f_2}{\leftarrow}A_2\). Algebras and homomorphisms can be described in an arbitrary way. We prove that for computing the fibred product it is enough to restrict to the case when $f_i,i=1,2$ are surjective and describe how to reduce to the surjective case. Also the way of choosing generators and relations for input algebras is considered. Paper is published in the author's wording.

Dynkin algebras are studied. Such algebras form a useful instrument for discussing probabilities in a rather natural context. Abstractness means the absence of a set-theoretic structure of elements in such algebras. A large useful class of abstract algebras, separable Dynkin algebras, is introduced, and the simplest example of a nonseparable algebra is given. Separability allows us to define appropriate variants of Boolean versions of the intersection and union operations on elements. In general, such operations are defined only partially. Some properties of separable algebras are proved and used to obtain the standard intersection and union properties, including associativity and distributivity, in the case where the corresponding operations are applicable. The established facts make it possible to define Boolean subalgebras in a separable Dynkin algebra and check the coincidence of the introduced version of the definition with the usual one. Finally, the main result about the structure of separable Dynkin algebras is formulated and proved: such algebras are represented as set-theoretic unions of maximal Boolean subalgebras. After preliminary preparation, the proof reduces to the application of Zorn’s lemma by the standard scheme.