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A latarre is a lattice with an arrow. its axiomatization looks natural. Latarres have a nontrivial theory which permits many constructions of latarres. Latarres appear as an end result of a series of generalizations of better known structures. These include Boolean algebras and Heyting algebras. Latarres need not have a distributive lattice. © 2017 Springer Science+Business Media B.V  

A latarre is a lattice with an arrow. Its axiomatization looks natural. Latarres have a nontrivial theory which permits many constructions of latarres. Latarres appear as an end result of a series of generalizations of better known structures. These include Boolean algebras and Heyting algebras. Latarres need not have a distributive lattice. © 2017, Springer Science+Business Media B.V  

We study the linear Lindenbaum algebra of Basic Propositional Calculus, called linear basic algebra. © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim  

The commuting graph of a ring R, denoted by Γ(R), is a graph of all whose vertices are noncentral elements of R, and 2 distinct vertices x and y are adjacent if and only if xy = yx. In this article we investigate some graph-theoretic properties of Γ (kG), where G is a finite group, k is a field, and 0 ≠ ⌊G⌋ j∈ k. Among other results it is shown that if G is a finite nonabelian group and k is an algebraically closed field, then GΓ (kG) is not connected if and only if ⌊G⌋ = 6 or 8. For an arbitrary field k, we prove that Γ (kG)is connected if G is a nonabelian finite simple group or G' ≠ G" and G"≠ 1  

We study Basic algebra, the algebraic structure associated with basic propositional calculus, and some of its natural extensions. Among other things, we prove the amalgamation property for the class of Basic algebras, faithful Basic algebras and linear faithful Basic algebras. We also show that a faithful theory has the interpolation property if and only if its correspondence class of algebras has the amalgamation property. © 2006 Springer-Verlag  

In this paper, we complement some recent results of L. Márki, J. Meyer, J. Szigeti and L. van Wyk, by investigating the constant-trace representations of a Clifford algebra C(V) of an arbitrary quadratic form q: V→ F (possibly degenerate) and we present some relevant applications. In particular, the existence of the polynomial identities of C(V) of particular form when the characteristic of the base field is zero is looked at. Furthermore, a lower bound is found on the minimal number t, such that C(V) can be embedded in a matrix ring of degree t, over some commutative F-algebra. Also, some results on the dimension of commutative subalgebras of C(V) are obtained. © 2021, The Author(s), under... 

Let R be a left artinian central F-algebra, T(R) = J(R) + [R,R], and U(R) the group of units of R. As one of our results, we show that, if R is algebraic and char F = 0, then the number of simple components of R = R/J(R) is greater than or equal to dimF R/T(R). We show that, when char F = 0 or F is uncountable, R is algebraic over F if and only if [R, R] is algebraic over F. As another approach, we prove that R is algebraic over F if and only if the derived subgroup of U(R) is algebraic over F. Also, we present an elementary proof for a special case of an old question due to Jacobson. © Inst. Math. CAS 2001  

We argue that one can relax the requirements of the non-associative three-algebras recently used in constructing D = 3, = 8 superconformal field theories, and introduce the notion of ''relaxed three-algebras''. We present a specific realization of the relaxed three-algebras in terms of classical Lie algebras with a matrix representation, endowed with a non-associative four-bracket structure which is prescribed to replace the three-brackets of the three-algebras. We show that both the so(4)-based solutions as well as the cases with non-positive definite metric find a uniform description in our setting. We discuss the implications of our four-bracket representation for the D = 3, = 8 and multi... 

We propose a general framework for deriving the OPEs within a logarithmic conformal field theory (LCFT). This naturally leads to the emergence of a logarithmetic partner of the energy momentum tensor within an LCFTand implies that the current algebra associated with an LCFT is expanded. We derive this algebra for a generic LCFTand discuss some of its implications. We observe that two constants arise in the OPE of the energy-momentum tensor with itself. One of these is the usual central charge  

We study the variety of Löb algebras, the algebraic structures associated with Formal Propositional Calculus. Among other things, we show that there exist only two maximal intermediate logics in the lattice of intermediate logics over Basic Propositional Calculus. We introduce countably many locally finite sub-varieties of the variety of Löb algebras and show that their corresponding intermediate logics have the interpolation property. Finally, we characterize all chain basic algebras with empty set of generators, and show that there are continuum many such chain basic algebras  

A necessary and sufficient condition for a central simple algebra with involution over a field of characteristic two to be decomposable as a tensor product of quaternion algebras with involution, in terms of its Frobenius subalgebras, is given. It is also proved that a bilinear Pfister form, recently introduced by A. Dolphin, can classify totally decomposable central simple algebras of orthogonal type  

We study the variety of Löb algebras, the algebraic structures associated with formal prepositional calculus. Among other things, we prove a completeness theorem for formal prepositional logic with respect to the variety of Löb algebras. We show that the variety of Löb algebras has the weak amalgamation property. Some interesting subclasses of the variety of Löb algebras, e. g. linear, faithful and strongly linear Löb algebras are introduced. © 2006 WILEY-VCH Verlag GmbH & Co. KGaA  

Involutions of the Clifford algebra of a quadratic space induced by orthogonal symmetries are investigated  

Given a non-commutative finite dimensional F-central division algebra D, we study conditions under which every non-abelian maximal subgroup M of GL n(D) contains a non-cyclic free subgroup. In general, it is shown that either M contains a non-cyclic free subgroup or there exists a unique maximal subfield K of Mn(D) such that NGLn<(D) (K*)=M, K* M, K/F is Galois with Gal(K/F) ≅ M/K*, and F[M]=Mn(D). In particular, when F is global or local, it is proved that if ([D:F],Char(F))=1, then every non-abelian maximal subgroup of GL 1(D) contains a non-cyclic free subgroup. Furthermore, it is also shown that GLn(F) contains no solvable maximal subgroups provided that F is local or global and n ≥ 5. ©... 

The aim of this article is to provide a characterization of quadratic forms of low dimension such that the canonical involutions of their Clifford algebras are hyperbolic  

Given an F-central simple algebra A = M n(D), denote by A′ the derived group of its unit group A*. Here, the Frattini subgroup Φ(A*) of A* for various fields F is investigated. For global fields, it is proved that when F is a real global field, then Φ(A*) = Φ(F*)Z(A′), otherwise Φ(A*) = ∩ pdoes not dividedeg(A) F *p. Furthermore, it is also shown that Φ(A*) = k* whenever F is either a field of rational functions over a divisible field k or a finitely generated extension of an algebraically closed field k  

This is a variation on a theme of Bayer-Fluckiger, Shapiro, and Tignol related to hyperbolic involutions. More precisely, criteria for the hyperbolicity of involutions of quadratic extensions of simple algebras and involutions of the form σ ⊗ τ and σ ⊗ ρ, where σ is an involution of a central simple algebra A, τ is the nontrivial automorphism of a quadratic extension of the center of A, and ρ is an involution of a quaternion algebra are obtained