Showing papers on "Idempotence published in 1968"

Commutative non-associative algebras and identities of degree four.

Abstract: The main result of this paper is the following. Theorem 1. Let A be a simple, commutative, finite-dimensional algebra containing an idempotent over a field of characteristic 0, and let the algebra A' obtained from A by adjoining a unity element satisfy an identity of degree ≦ 4 not implied by commutativity. Then either A is a Jordan algebra or A is two-dimensional over an appropriate field E.

A note on reduced jordan algebras

Abstract: In this paper we give short proofs of two of the main theorems concerning reduced exceptional Jordan algebras A = H(C3, 'y): the AlbertJacobson Theorem that the Cayley coordinate algebra C is determined by A up to isomorphism, and the Springer Theorem that two such algebras A, A' are isomorphic if and only if they have isomorphic coordinate algebras and equivalent trace forms. We avoid using the generic norm by working directly with the reduced idempotents. Our proofs do not require that the algebras be exceptional, and are valid for arbitrary reduced simple algebras. 1. Reduced idempotents. Throughout this paper, A will denote a Jordan algebra with identity element 1 over a field 1 of characteristic 5 2. In this section we make no assumptions about the simplicity or finite-dimensionality of A. Recall that an idempotent e is reduced if Ai(e) = UeA =cJe. We assume the reader is familiar with the operators U, = 2L' -Lx2and the basic identities involving them (e.g. [3,

A description of all globally idempotent threads with zero

Abstract: The term \"thread\" was introduced by A. H. Clifford in [1] to designate a connected topological semigroup in which the topology is that induced by a total order relation. A thread S is said to be globally idempotent if S2 = S. In [6] the author has shown that, after reversing the order if necessary, the subset {x | 0 á x} in a globally idempotent thread with zero is a subthread having a particularly pleasant structure. This result is the foundation on which the description given in this paper is based. An analogous situation existed in the characterization given by Cohen and Wade [4] of all topological semigroups on a compact real interval which have a zero and an identity. In such a thread, the identity must be an endpoint, and the closed interval between the zero and the identity a subthread. Assuming the identity to be the maximal element, this subthread is then, in the terminology of [1], a standard thread. Since a characterization of all standard threads on a real interval had previously been given by Mostert and Shields [5], the problem solved by Cohen and Wade was also that of utilizing a given structure theorem for {x | O^x} to formulate a description of the whole thread. Consequently, many of the ideas developed by Cohen and Wade have again been used here. Treating the same type of problem, Clifford determined in fl] all possible compact threads having a zero and idempotent endpoints, and again, some of our results are simple generalizations of those in [1]. Considering the work of Clifford and of Cohen and Wade, the contribution of the present investigation toward a description of all globally idempotent threads with zero lies in dropping the requirement of compactness and in replacing the assumption of idempotent endpoints or of an identity by the weaker one of global idempotency. The terminology and notation will be essentially the same as that used in [6]. In particular, a standard thread is a compact thread in which the least element is a zero and the largest is an identity. We note that the trivial thread consisting of a zero alone is considered a standard thread. A thread with a zero and an identity