Showing papers on "Idempotence published in 1980"

The semigroup of singular endomorphisms of a finite dimensional vector space

Abstract: J.A. Erdos [1967b] proved that the semigroup Sing n of singular endomorphisms of an n -dimensional vector space V is generated by the set E of idempotent endomorphisms of rank n − 1. His proof depended entirely on matrix theory and shed very little light on the structure of the semigroup. The object of the first half of this talk is to give a more illuminating proof of this result. From the proof given by Erdos it could be deduced that any element of Sing n could be expressed as the product of 2 n elements of E . The second half of this talk will be devoted to reducing this bound to n , which can then be shown to be best possible.

Computations in choice groupoids

Abstract: Defining achoice as a mapping of the subsets of a setX into their respective subsets, a one-to-one (and “naturally”) corresponding binary operation,sequential choice, is identified under which the power set ofX is closed as achoice groupoid. A complete logical diagram is given, exhibiting all the implications between conjunctions of the seven conditions: (1) idempotence, (2) “consistency,” (3) “absorbence,” and (4) homomorphism of a choice, and (5) commutativity, (6) associativity, and (7) “path-independence” of the corresponding sequential choice.

Alternative rings whose symmetric elements are nilpotent or a right multiple is a symmetric idempotent.

Abstract: Osborn characterizes those associative rings with involu- tions in which each symmetric element is nilpotent or invertible. Analogous results are obtained for alternative rings. The restriction is further relaxed to require only that each symmetric element is nilpotent or some multiple is a symmetric idempotent. Introduction* J. M. Osborn (10) (11) proved a series of theo- rems concerning the structure of associative rings with involution such that any symmetric element is either nilpotent or invertible. Many generalizations of his results have appeared in the literature for associative rings (a good single reference is Herstein (4)). We begin with a discussion of involutions in the Cayley-Dickson algebras. Then Osborn's results are generalized to alternative rings. Our final result shows that if R is an alternative ring with involution * such that (a) each symmetric element s is either nilpotent or some right multiple of s is a symmetric idempotent and (b) each set of pairwise orthogonal symmetric idempotent has n or less elements, then the quotient ring i?/RadJ? has d.c.c on right ideals. Since a radical free alternative ring with d.c.c. on right ideals is the direct sum of Cayley-Dickso n algebras and simple artinian associative rings, we have a nice description of these quotient rings.

Maximal Elements in the Maximal Ideal Space of a Measure Algebra

Abstract: Let ~ ' be a semi-simple commutative convolution measure algebra as described by Taylor [9]. The maximal ideal space A of ~ ' is representable as the semigroup of continuous semicharacters on a compact semigroup the structure semigroup of J/g. Using this description of A, Taylor has shown that the strong boundary points he A of ~ ' must have idempotent modulus (Ihl 2 = 1hi). On the other hand it is relatively easy to exhibit idempotents which are not strong boundary points in fact, which are not even in the Silov boundary. For example, let 2 /cons i s t of all summable sequences (a,),__> o with convolution as multiplication. Then the idempotent complex homomorphism (a~),,~a o is not in the Silov boundary of ~ ' . For the most important example, where ~ ' =M(G), the convolution algebra of finite regular complex valued Borel measures on a locally compact abelian group G, the question of whether idempotents are strong boundary points is less trivial. In fact the question of whether all idempotents are in the Silov boundary of M(G) remains unresolved. However, it is known (see [3]) that if the idempotent h in question corresponds to the direct sum decomposition of M(G) produced by a single generator symmetric Raikov system then there is an analytic disc around h and this can be used to show that h is not a strong boundary point. To be precise the disc is analytic not only for M(G) but for the uniform closure M(G~'-of the algebra of Gelfand transforms on A, and this prevents h being a strong boundary point. As a simple example, if the Raikov system consists of all countable subsets of G, then h is the complex homomorphism/~,~ S d#n where #n is the discrete part