Topic
Idempotence
About: Idempotence is a research topic. Over the lifetime, 1860 publications have been published within this topic receiving 19976 citations. The topic is also known as: idempotent.
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TL;DR: In this article, it was shown that a mode (idempotent and entropic algebra) is a Lallement sum of its cancellative submodes over a normal band if it has a congruence with the normal band quotient and cancellative classes.
Abstract: A mode (idempotent and entropic algebra) is a Lallement sum of its cancellative submodes over a normal band if it has a congruence with a normal band quotient and cancellative congruence classes. We show that such a sum embeds as a subreduct into a semimodule over a certain ring, and discuss some consequences of this fact. The result generalizes a similar earlier result of the authors proved in the case when the normal band is a semilattice.
6 citations
TL;DR: In this paper, it was shown that any unitary commutative locally convex algebra with a continuous product which is a Baire space and in which all entire functions operate is actually m-convex.
Abstract: We show that a unitary commutative locally convex algebra, with a continuous product which is a Baire space and in which entire functions operate is actually m-convex. Whence, as a consequence, the same result of Mitiagin, Rolewicz and Zelazko, in commutative B0-algebras. It is known that entire functions operate in complete m-convex algebras [1]. In [3] Mitiagin, Rolewicz and Zelazko show that a unitary commutative B0-algebra in which all entire functions operate is necessarily m-convex. Their proof is quite long and more or less technical. They use particular properties of B0-algebras, a Baire argument and the polarisation formula. Here we show that any unitary commutative locally convex algebra, with a continuous product which is a Baire space and in which all entire functions operate is actually m-convex. The proof is short, direct and selfcontained. A locally convex algebra (A; ), l. c. a. in brief, is an algebra over a eld K (K = R or C) with a Hausdor locally-convex topology for which the product is separately continuous. If the product is continuous in two variables, (A; ) is said to be with continuous product. A l. c. a. (A; ) is said to bem-convex (l. m. c. a.) if the origin 0 admits a fundamental system of idempotent neighbourhoods ([2]). An
6 citations
TL;DR: For any positive integer n, a necessary and sufficient condition for an n × n r-circulant over the Boolean algebra B = {0, 1} to be idempotent was given in this paper.
6 citations
6 citations
TL;DR: In this article, the authors give bounds on the distance from a non-zero idempotent to the set of nilpotents in a set of n × n matrices.
Abstract: We give bounds on the distance from a non-zero idempotent to the set of nilpotents in the set of n × n matrices in terms of the norm of the idempotent. We construct explicit idempotents and nilpotents which achieve these distances, and determine exact distances in some special cases.
6 citations