Necessary subalgebras of simple nonintegral semiassociative relation algebras

There are six finite nonintegral representable relation algebras such that every nonintegral simple semiassociative relation algebra has a nontrivial subalgebra isomorphic to one of those six.Presented by Bjarni Jónsson.     

The orthogonality relation classifies finite-dimensional formally real simple Jordan algebras

Abstract

It is shown that a finite-dimensional formally real simple Jordan algebra is completely determined by the relation of Jordan-orthogonality.

Communicated by Prof. Alberto Elduque     

Central simple algebras

Wedderburn’s factorization of polynomials over division rings is refined and used to prove that every central division algebra of degree 8, with involution, has a maximal subfield which is a Galois extension of the center (with Galois group Z₂⊕Z₂⊕Z₂). The same proof, for an arbitrary central division algebra of degree 4, gives an explicit construction of a maximal subfield which is a Galois extension of the center, with Galois group Z₂⊕Z₂. Use is made of the generic division algebras, with and without involution. This work was supported by the Israel Committee for Basic Research and the Anshel Pfeffer Chair.     

Matrix relation algebras

Square matrices over a relation algebra are relation algebras in a natural way. We show that for fixed n, these algebras can be characterized as reducts of some richer kind of algebra. Hence for fixed n, the class of n × n matrix relation algebras has a first–order characterization. As a consequence, homomorphic images and proper extensions of matrix relation algebras are isomorphic to matrix relation algebras. Received July 18, 2001; accepted in final form April 24, 2002.     

Bifunctional-elementary relation algebras

A relation algebra is bifunctional-elementary if it is atomic and for any atom a, the element a;1;a is the join of at most two atoms, and one of these atoms is bifunctional (an element x is bifunctional if ’). We show that bifunctional-elementary relation algebras are representable. Our proof combines the representation theorems for: pair-dense relation algebras given by R. Maddux; relation algebras generated by equivalence elements provided corresponding relativizations are representable by S. Givant; and strong-elementary relation algebras dealt with in our earlier work. It turns out that atomic pair-dense relation algebras are bifunctional elementary, showing that our theorem generalizes the representation theorem of atomic pair-dense relation algebras. The problem is still open whether the related classes of rather elementary, functional-elementary, and strong functional-elementary relation algebras are representable. Received July 15, 2007; accepted in final form March 17, 2008.     

Domestic canonical algebras and simple Lie algebras

For each simply-laced Dynkin graph Δ we realize the simple complex Lie algebra of type Δ as a quotient algebra of the complex degenerate composition Lie algebra of a domestic canonical algebra A of type Δ by some ideal I of that is defined via the Hall algebra of A, and give an explicit form of I. Moreover, we show that each root space of has a basis given by the coset of an indecomposable A-module M with root easily computed by the dimension vector of M. Dedicated to Professor Claus Michael Ringel on the occasion of his 60th birthday.     

Minimal relation algebras

This paper is concerned with the covers of the atoms in the lattice of varieties of relation algebras. Aminimal relation algebra is one that is simple and generates such a subvariety. The main result we prove is that there are exactly three finite minimal relation algebras that aretotally symmetric (i.e., satisfy the identitiesx=x andx x; x). We also give an example of an infinite minimal totally symmetric relation algebra, and some results about other subvarieties.Dedicated to Bjarni Jonsson on his 70th birthdayPresented by G. McNulty.     

Coset relation algebras

A measurable relation algebra is a relation algebra in which the identity element is a sum of atoms that can be measured in the sense that the “size” of each such atom can be defined in an intuitive and reasonable way (within the framework of the first-order theory of relation algebras). A large class of examples of such algebras, using systems of groups and coordinated systems of isomorphisms between quotients of the groups, has been constructed. This class of group relation algebras is not large enough to exhaust the class of all measurable relation algebras. In the present article, the class of examples of measurable relation algebras is considerably extended by adding one more ingredient to the mix: systems of cosets that are used to “shift” the operation of relative multiplication. It is shown that, under certain additional hypotheses on the system of cosets, each such coset relation algebra with a shifted operation of relative multiplication is an example of a measurable relation algebra. We also show that the class of coset relation algebras does contain examples that are not representable as set relation algebras. In later articles, it is shown that the class of coset relation algebras is adequate to the task of describing all measurable relation algebras in the sense that every atomic measurable relation algebra is essentially isomorphic to a coset relation algebra, and the class of group relation algebras is similarly adequate to the task of representing all measurable relation algebras in which the associated groups are finite and cyclic.     

Relativized relation algebras

Relativization is one of the central topics in the study of algebras of relations (i.e. relation and cylindric algebras). Relativized representable relation algebras behave much nicer than the original class RAA: for instance, one obtains finite axiomatizability, decidability and amalgamation by relativization. The properties of the class obtained by relativizing RRA depend on the kind of element with which the algebras are relativized. We give a systematic account of all interesting choices of relativizing RRA, and show that relativizing with transitive elements forms the borderline where all above mentioned three properties switch from negative to positive. Received January 24, 1993; accepted in final form October 7, 1998.     

Many-valued relation algebras

We introduce MV-relation algebras (MVRAs) and distributive MV-relation algebras (DMVRAs), many-valued generalizations of classical relation algebras and study some of their arithmetical properties. We provide corresponding notions of group relation algebra and complex algebra and generalize some results about them from the classical case. For this, we work with more general structures than MVRAs and DMVRAs, by replacing the MV part with a BL-algebra, obtaining what we call fuzzy relation algebras and distributive fuzzy relation algebras.Dedicated to the Memory of Wim BlokReceived February 12, 2003; accepted in final form October 15, 2004.This revised version was published online in August 2005 with a corrected cover date.     

A simple construction of representable relation algebras with non‐representable completions

We give a simple new construction of representable relation algebras with non‐representable completions. Using variations on our construction, we show that the elementary closure of the class of completely representable relation algebras is not finitely axiomatizable (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogeneously simple associative algebras

We prove that every finite-dimensional homogeneously simple associative algebra over an algebraically closed field is representable as the product of a full matrix algebra and a graded field.     

Differentiably simple alternative algebras

Differentiably simple alternative nonassociative algebras of characteristic p > 0 are described in terms of differentiably simple associative commutative algebras. Also we look at some properties of differentiably simple alternative nonassociative algebras of characteristic 0.     

Differentiably simple Jordan algebras

We prove that each exceptional differentiably simple Jordan algebra over a field of characteristic 0 is an Albert ring whose elements satisfy a cubic equation with the coefficients in the center of the algebra. If the characteristic of the field is greater than 2 then such an algebra is the tensor product of its center and a central exceptional simple 27-dimensional Jordan algebra. Some remarks made on special algebras.     

Invariants of simple algebras

We determine the group of invariants with values in Galois cohomology with coefficients of central simple algebras of degree at most 8 and exponent dividing 2. The work of A. S. Merkurjev has been supported by the NSF grant DMS #0652316.     

Central simple Poisson algebras

Poisson algebras are fundamental algebraic structures in physics and sym-plectic geometry. However, the structure theory of Poisson algebras has not been well developed. In this paper, we determine the structure of the central simple Poisson algebras related to locally finite derivations, over an algebraically closed field of characteristic zero. The Lie algebra structures of these Poisson algebras are in general not finitely-graded.