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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

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

We study the properties and relations of fuzzy pseudo-filters of pseudo-BCK algebras. After we discuss the equivalent conditions of fuzzy normal pseudo-filter of pseudo-BCK algebra (pP), we propose fuzzy implicative pseudo-filter and its relation with fuzzy Boolean filter of (bounded) pseudo-BCK algebras (pP). Then two open problems: "In pseudo-BCK algebra or bounded pseudo-BCK algebra, Is the notion of implicative pseudo-filter equivalent to the notion of Boolean filter?" and "Prove or negate the following conclusion: A pseudo-BCK algebra is an implicative pseudo-BCK algebra if and only if every pseudo-filter of it is a Boolean filter(or an implicative pseudo-filter)" are partly solved.

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.

We present Poisson algebra constructions on the base of associative algebras and study the relationship between the obtained Poisson algebras and the initial associative algebras. Based on obtained constructions, we present two classes of minimal varieties of Poisson algebras of polynomial growth, i.e., varieties for which the sequence of codimensions grows as a polynomial of degree k, but the sequence of codimensions of any proper subvariety grows as a polynomial whose degree is strictly less than k. Previously the author showed that there are only two varieties of Poisson algebras of almost polynomial growth. In this paper we give a complete description of all subvarieties of these two varieties.

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.

In this paper, we show that every weakly algebraic ideal of an effect algebra E induces a uniform topology (weakly algebraic ideal topology, for short) with which E is a first-countable, zero-dimensional, disconnected, locally compact and completely regular topological space, and the operation ⊕ of effect algebras is continuous with respect to these topologies. In addition, we prove that the operation ⊝ of effect algebras and the operations ∧ and ∨ of lattice effect algebras are continuous with respect to the weakly algebraic ideal topology generated by a Riesz ideal.

We study equations over boolean algebras with distinguished elements. We prove the criteria, when a boolean algebra is equationally Noetherian, weakly equationally Noetherian, $\mathbf{q}_\omega$-compact or $\mathbf{u}_\omega$-compact. Also we solve the problem of geometric equivalence in the class of boolean algebras with distinguished elements.

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.

We develop a theory of ordered *-algebras with an order unit. These are complex algebras equipped with a conjugate-linear vector space involution whose hermitian elements form an ordered real algebra with an order unit but not necessarily with a multiplicative identity. Our main result is a representation theorem for ordered *-algebras with an order unit. This can be viewed as the complex and the non-unital version of the representation theorem of Stone and Kadison for ordered real algebras containing an order unit which is a multiplicative identity. A key role in our approach is played by certain weighted function algebras.

In Bezhanishvili et al. (2012) we introduced the category MKHaus of modal compact Hausdorff spaces, and showed these were concrete realizations of coalgebras for the Vietoris functor on compact Hausdorff spaces, much as modal spaces are coalgebras for the Vietoris functor on Stone spaces. Also in Bezhanishvili et al. (2012) we introduced the categories MKRFrm and MDV of modal compact regular frames, and modal de Vries algebras as algebraic counterparts to modal compact Hausdorff spaces, much as modal algebras are algebraic counterparts to modal spaces. In Bezhanishvili et al. (2012), MKRFrm and MDV were shown to be dually equivalent to MKHaus, hence equivalent to one another. Here we provide a direct, choice-free proof of the equivalence of MKRFrm and MDV. We also detail connections between modal compact regular frames and the Vietoris construction for frames (Johnstone 1982, 1985), discuss a Vietoris construction for de Vries algebras, and how it is linked to modal de Vries algebras. Also described is an alternative approach to the duality of MKRFrm and MKHaus obtained by using modal de Vries algebras as an intermediary.

This study determines all third power-associative (-1, 1) algebras, whose commutator algebras are generalized Kac-Moody algebras.

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.

In the paper we find characterizations of some notions studied by Kułaga [5] and we generalize his results. In particular, we characterize regularity and completeness of factor subalgebras via stability of the decidability operator and we discuss some possibilities in defining the notions of first category and Baire property in Boolean algebras.

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.

A topological algebra A is called a Q-algebra if its set of quasi-invertible elements (of invertible elements, if A is unital) is an open set. This class of topological algebras has been extensively studied by several authors, but mostly in unital or commutative case. Properties of (not necessarily unital or commutative) TQ-algebras and, in the special case, of TQ-algebras with functional topological spectrum are given in the present paper.

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

In this work, we focus on the study of the algebra of strong subobjects obtained from a category of rough sets that forms a quasitopos. A new algebraic structure called ‘contrapositionally complemented pseudo Boolean algebra’ is obtained and its basic properties studied. The corresponding logic ‘intuitionistic logic with minimal negation’ is introduced, and its connection with the intuitionistic and minimal logics is discussed.

A relation algebra is said to be minimal if it is generated by the empty set, or equivalently, if it is generated by the identity element. These algebras were already discussed in Section 6.7 in the context of minimal subalgebras; however, that discussion was necessarily incomplete because the tools required for a systematic analysis of these algebras were not yet available.

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.

Effect algebras (EAs), introduced by D. J. Foulis and M. K. Bennett, as common generalizations of Boolean algebras, orthomodular lattices and MV-algebras, are nondistributive algebraic structures including unsharp elements. Their unbounded versions, called generalized effect algebras, are posets which may have or may have not an EA-MacNeille completion, or cannot be embedded into any complete effect algebra. We give a necessary and sufficient condition for a generalized effect algebra to have an EA-MacNeille completion. Some examples are provided.

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.

Algebras of imperative programming languages have been successful in reasoning about programs. In general an algebra of programs is an algebraic structure with programs as elements and with program compositions (sequential composition, choice, skip) as algebra operations. Various versions of these algebras were introduced to model partial correctness, total correctness, refinement, demonic choice, and other aspects. We introduce here an algebra which can be used to model total correctness, refinement, demonic and angelic choice. The basic model of our algebra are monotonic Boolean transformers (monotonic functions from a Boolean algebra to itself).

The paper continues the previous author’s paper in which the bimodule resolution was constructed for algebras of semidihedral type from the family SD(2ℬ)2. In the present paper, using this resolution the multiplicative structure of the Hochschild cohomology algebra for algebras in the mentioned family is described over a base field of characteristic 2 under an additional assumption that the parameter c contained in the defining relations of the algebras is equal to zero.

Commutative vertex algebras arising as subalgebras of the vertex algebras corresponding to the Kac-Moody algebras are studied. Systems of defining relations and degenerations into algebras with quadratic relations are described. The results can be used to obtain fermionic formulas for characters.

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.

In this paper, we introduce the definition of generalized H-Hom–Lie algebras (i.e., Hom-Lie algebras in the Yetter–Drinfeld category ) for any Hopf algebra H, and obtain generalized H-Hom-Lie algebras from H-Hom-associative algebras (i.e., Hom-associative algebras in ) and generalized H-Lie algebras (i.e., Lie algebras in ), respectively. We show that if A is a sum of two H-commutative Hom-associative subalgebras, then the H-commutator Hom-ideal of A is nilpotent. Finally, we describe the H-Hom-Lie ideal structures of the H-Hom-associative algebras by analogy with that of H-algebras.

A function is algebraic on an algebra if it can be implicitly defined by a system of equations on . In this note we give a semantic characterization for algebraic functions on quasiprimal algebras. This characterization is applied to obtain necessary and sufficient conditions for a quasiprimal algebra to have every one of its algebraic functions be a term function. We also apply our results to particular algebras such as finite fields and monadic algebras.

Research into universal algebras (for different classifications included) is generally confined to working with termal (or polynomial) functions of these algebras. Attempts to go beyond this scope while staying within the frames of functions naturally definable on the algebras under consideration led the author to the idea of studying conditional termal functions (and their different generalizations such as positively, elementarily conditional termal, implicit, and abstract functions). As a continuation of studies in naturally definable functions on universal algebras, we propose to consider L-definable functions, where L is some logical language. This most general approach turns out to be connected with a scheme for defining conditional termal functions and their generalizations, as well as with various derivative structures of universal algebras. Here we present L-definable functions on universal algebras and outline their basic properties. On this basis, also, we introduce the notion of L-definably equivalent algebras, which is a generalization of the concept of being rationally equivalent for algebras.

We give a complete description of conditions of being strongly constructivizable for Boolean algebras of elementary characteristic (∞, 0, 0) in terms of being computable for a sequence of canonical Ershov–Tarski predicates on Boolean algebras.

We study the so-called pairwise summable generalized effect algebras and show some conditions under which they are generalized MV-effect algebras. Moreover, we give a necessary and sufficient condition for finite antichains of elements in pairwise summable generalized effect algebras under which they are sets of atoms of sub-generalized effect algebras, which are generalized MV-effect algebras in their own right. Simultaneously, we give counterexamples which show properties of those antichains which are not necessary or sufficient to be atoms of such generalized MV-effect algebras.