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

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.

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.

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 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

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...

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.

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.

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.

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.

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.

In 2007, Maddux observed that certain classes of representable relationalgebras (RRAs) form sound semantics for some relevant logics. In particular,(a) RRAs of transitive relations are sound for R, and (b) RRAs of transitive,dense relations are sound for RM. He asked whether they were complete aswell. Later that year I proved a modest positive result in a similar direction,namely that weakly associative relation algebras, a class (much) larger thanRRA, is sound and complete for positive relevant logic B. In 2008, Mikulas proved anegative result: that RRAs of transitive relations are not complete for R.His proof is indirect: he shows that the quasivariety of appropriatereducts of transitive RRAs is not finitely based. Later Maddux re-establishedthe result in a more direct way. In 2010, Maddux proved a contrasting positive result: that transitive, dense RRAs are complete for RM. He found an embedding of Sugihara algebras into transitive, dense RRAs. I will show that if we give up the requirement of representability, the positive result holds for R as well. To be precise, the following theorem holds.Theorem. Every normal subdirectly irreducible De Morgan monoid in the languagewithout Ackermann constant can be embedded into a square-increasing relationalgebra. Therefore, the variety of such algebras is sound and complete for R.

Residuated structures are important lattice-ordered algebras both for mathematics and for logics; in particular, the development of lattice-valued mathematics and related non-classical logics is based on a multitude of lattice-ordered structures that suit for many-valued reasoning under uncertainty and vagueness. Extended-order algebras, introduced in [10] and further developed in [1], give an order-theoretical approach to a general description of the algebras of logics which goes along the line of an implication-based view. In the metamathematical framework of classical and intuitionistic logics the algebraic structure of their semantics depends entirely on an order relation in the set of the truth values. In fact, the algebraic structure of both boolean and Heyting algebras (or, as one should prefer saying, boolean and Heyting lattices) is completely determined by the underlying order relation. Looking at non-classical logics, extended-order algebras have been introduced on the base of the following principle: "just like an order relation in a set L determines completely the lattice structure of L, each of its extensions relative to a true value>2 L, e.g. any implication !: L L! L such that for all a;b2 L : a b , a! b =>, completely determines the richer lattice-ordered algebraic structure on L, to be used either in classical or in non-classical logics". The implicative structure (L;!;>) so obtained has been called implicative algebra in the monograph of H. Rasiowa [12], where it has been specialized to characterize either algebras of subsets (implication algebras) or algebras of open sets (positive implication algebras also called Hilbert algebras). Instead, the above described motivation has led to call (L;!;>) weak extended-order algebra (w-eo algebra) in [10, 1]; there, additional conditions have been considered and discussed, including a weak, but important requirement that characterizes extended-order algebras (eo algebras), giving characterizations of several classes of residuated structures; in particular, it is seen that every integral residuated lattice is a symmetrical distributive and associative eo algebra. It has to be noted that

Alternative just infinite-dimensional algebras are studied, i.e., infinite-dimensional algebras in which every nonzero ideal has finite codimension. It is proved that these algebras are prime. In the nonassociative case, the Noetherian property with respect to one-sided ideals is proved, and the cases of Cayley–Dickson rings and exceptional algebras are investigated.

Algebras having bases that consist entirely of units (called invertible algebras) are studied. Among other results, it is shown that all finite-dimensional algebras over a field other than the binary field F2 have this property. Invertible finite-dimensional algebras over F2 are fully characterized. Examples of invertible algebras are shown to include all (non-trivial) matrix rings over arbitrary algebras. In addition, various families of algebras, including group rings and crossed products, are characterized in terms of invertibility. Invertibility of infinite-dimensional algebras is also explored.

A general construction of the free algebra over a poset in varieties finitely generated is given in [8]. In this paper, we apply this to the varieties of Łukasiewicz-Moisil algebras, giving a detailed description of the free algebra over a finite poset (X, ≤) , Freen((X, ≤)). As a consequence of this description, the cardinality of Freen((X, ≤)). is computed for special posets.

We construct cubic Jordan algebras over an integral proper scheme X such that 2, 3 ∈ H 0(X, 𝒪 X ), generalizing a construction by B. N. Allison and J. R. Faulkner. In the process, we obtain admissible cubic algebras and pseudocomposition algebras over X. Results on the structure of these algebras are obtained, as well as examples over elliptic curves.

The compact information algebra and the continuous information algebra are two special information algebras, which are algebraic structures modeling computation in many different contexts. We show that the set of all continuous mappings between two continuous information algebras also forms a continuous information algebra. Further, we obtain that the two categories COMP and CON, consisting of compact information algebras and continuous information algebras as objects respectively, continuous mappings as morphisms, are both Cartesian closed. We present an equivalent characterization of continuous information algebras.The set of all continuous mappings on continuous information algebras is continuous.The two categories COMP and CON are both Cartesian closed.

Relation algebras give rise to partial algebras on maps, which are generalized to partial algebras on polymaps while preserving the properties of relation union and composition. A polymap is defined as a map with every point in the domain associated with a special set of maps. Polymaps can be represented as small subcategories of Set*, the category of pointed sets. Map composition and the counterpart of relation union for maps are generalized to polymap composition and sum. Algebraic structures and categories of polymaps are investigated. Polymaps present the unique perspective of an algebra that can retain many of its properties when its elements (maps) are augmented with collections of other elements.

In this study, we present techniques for constructing a lattice-ordered effect algebra using a given family of MV-effect algebras. First, we introduce the Greechie diagrams of the pastings of a family of MV-effect algebras. As application of the Greechie diagrams, we provide some sufficient conditions for pasting a lattice-ordered effect algebra using a family of MV-effect algebras. Especially, we illustrate a kind of technique of how to obtain a lattice-ordered effect algebra by substituting the atoms of an orthomodular lattice with the lattice-ordered effect algebras. Finally, we prove a version of the "Loop Lemma" for MV-effect algebras pasting.

In this article, we introduce a class of generalized matrix algebras, in which each algebra is called a normally upper triangular gm algebra, and characterise Gorenstein projective modules over this class of algebras. Moreover, a sufficient condition of strongly Gorenstein projective modules over normally upper triangular gm algebras is given. The importance of normally upper triangular gm algebras for us is that it includes the so-called path algebras of quivers over algebras and generalized path algebras. Due to this, we characterize Gorenstein projective modules and strongly Gorenstein projective modules over path algebras of quivers over algebras and generalized path algebras as applications of the main results on normally upper triangular gm algebras. At last, we give an example to show how all indecomposable Gorenstein projective modules over a given algebra are constructed by the result on generalized path algebras.

One of natural combinations of Kripke complete modal logics is the product, an operation that has been extensively investigated over the last 15 years. In this paper we consider its analogue for arbitrary modal logics: to this end, we use product-like constructions on general frames and modal algebras. This operation was first introduced by Y. Hasimoto in 2000; however, his paper remained unnoticed until recently. In the present paper we quote some important Hasimoto's results, and reconstruct the product operation in an algebraic setting: the Boolean part of the resulting modal algebra is exactly the tensor product of original algebras (regarded as Boolean rings). Also, we propose a filtration technique for Kripke models based on tensor products and obtain some decidability results.

WOS: 000412021500020, Codes over the finite sub-Hopf algebras A (n) of the (mod 2) Steenrod algebra A were studied by Dougherty and Vergili. In this paper we study some Euclidean and Hermitian self-dual codes over A (n) by considering Milnor basis elements.

International audience; We study the relationship of backcrossing algebras with mutation algebras and algebras satisfying ω-polynomial identities: we show that in a backcrossing algebra every element of weight 1 generates a mutation algebra and that for any polynomial identity f there is a backcrossing algebra satisfying f. We give a criterion for the existence of idempotent in the case of baric algebras satisfying a nonhomogeneous polynomial identity and containing a backcrossing subalgebra. We give numerous genetic interpretations of the algebraic results.

We study a notion of order in Jordan algebras based on the version for Jordan algebras of the ideas of Fountain and Gould as adapted to the Jordan context by Fern\'{a}ndez-L\'{o}pez and Garc\'{\i}a-Rus, making use of results on general algebras of quotients of Jordan algebras. In particular, we characterize the set of Lesieur-Croisot elements of a nondegenerate Jordan algebra as those elements of the Jordan algebra lying in the socle of its maximal algebra of quotients, and apply this relationship to extend to quadratic Jordan algebras the results of Fern\'{a}ndez-L\'{o}pez and Garc\'{\i}a-Rus on local orders in nondegenerate Jordan algebras satisfying the descending chain condition on principal inner ideals and not containing ideals which are nonartinian quadratic factors., Comment: 44 pages

We survey results on multiserial algebras, special multiserial algebras and Brauer configuration algebras. A structural property of modules over a special multiserial algebra is presented. Almost gentle algebras are introduced and we describe some results related to this class of algebras. We also report on the structure of radical cubed zero symmetric algebras.

Interesting and important examples of relation algebras may be constructed in a variety of ways: from sets of binary relations, from sets of Boolean matrices, from sets of formulas, from Boolean algebras, from groups, from projective geometries, from modular lattices, and from relatively small abstract algebraic structures. In this chapter we shall look at some of these constructions.

3-Lie algebras are constructed by commutative associative algebras, involutions and derivations. Then the 3-Lie algebras are obtained from Lie algebras and linear functions, and from group algebras F[G] of an abelian group G and homomorphisms . At the end of the paper, the 3-Lie algebras are obtained from Laurent polynomials , and whose structures are studied.

We describe algebras in which the centralizer of every non-scalar element is equal to the subalgebra generated by this element, and finite-dimensional algebras (over perfect fields) in which the centralizer of every non-scalar element is commutative.

We prove that the variety of Lie algebras arising from splicing operation coincides with the variety CM of centreby-metabelian Lie algebras. Using these Lie algebras we find the minimal dimension algebras generated the variety CM and the variety of its associative envelope algebras. We study the splicing n-ary operation. We show that all n-ary (n > 2) commutator algebras arising from this operation are nilpotent of index 3. We investigate the generalization of the splicing n-ary operation, and we formulate a series of open problems.

The theory of fuzzy implication algebras was proposed by Professor Wangming Wu in 1990. The present paper reviews the following two aspects of studies on FI-algebras: concepts, properties and some subclasses of FI-algebras; axiomatization of the class of FI-algebras and some of its important subclasses. The main results are summarized in the current paper, the relationships between FI-algebras and several classes of important fuzzy algebras are discussed, such as BL-algebras, MTL-algebras, and residuated lattices, and propositional calculus systems of several special classes of FI-algebras are shown.

We study finite-dimensional commutative algebras, which satisfy the Jacobi identity. Such algebras are Jordan algebras. We describe some of their properties and give a classification in dimensions n 7 over algebraically closed fields of characteristic not 2 or 3.

The aim of this paper is to present several techniques of constructing a lattice-ordered effect algebra from a given family of lattice-ordered effect algebras, and to study the structure of finite lattice-ordered effect algebras. Firstly, we prove that any finite MV-effect algebra can be obtained by substituting the atoms of some Boolean algebra by linear MV-effect algebras. Then some conditions which can guarantee that the pasting of a family of effect algebras is an effect algebra are provided. At last, we prove that any finite lattice-ordered effect algebra E without atoms of type 2 can be obtained by substituting the atoms of some orthomodular lattice by linear MV-effect algebras. Furthermore, we give a way how to paste a lattice-ordered effect algebra from the family of MV-effect algebras.

It is known that the variety of commutative basic algebras, the variety of MV-algebras, the variety of orthomodular lattices and hence also the variety of Boolean algebras are proper subvarieties of the variety of basic algebras. In the paper several possible axiomatizations of the variety of basic algebras are provided. The variety of symmetric basic algebras is introduced and shown to be a proper subvariety of the variety of commutative basic algebras, and whose members have the underlying lattices distributive. It is proved that commutative basic algebras can be described as residuated $$l$$ l -groupoids.

The (n + 1)-ary derivations of simple non-Lie n-ary Maltsev algebras are described in the case of the binary algebra M7 and the ternary algebra M8 .A s a consequence, a description is obtained for the 3-ary derivations of simple non-Lie Maltsev algebras and simple finite-dimensional non-Lie binary-Lie algebras. Exam- ples of semisimple Maltsev algebras having nontrivial 3-ary derivations are given.