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: For any element a in an exchange ring R, it is shown in this paper that there is an idempotent e ∈ Ra such that 1−e∈(1−a) R. The main theorem of this paper is a general two-sided statement on exchange elements in arbitrary rings which subsumes both of these results.
Abstract: For any element a in an exchange ring R, we show that there is an idempotent \(\,e\in aR\cap R\,a\,\) such that \(\,1-e\in (1-a)\,R\cap R\,(1-a)\). A closely related result is that a ring R is an exchange ring if and only if, for every a∈R, there exists an idempotent e∈Ra such that 1−e∈(1−a) R. The Main Theorem of this paper is a general two-sided statement on exchange elements in arbitrary rings which subsumes both of these results. Finally, applications of these results are given to the study of the endomorphism rings of exchange modules.
8 citations
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07 Jul 1997TL;DR: It is proved that the recognizable series are certain rational power series, which can be constructed from the polynomials by using the operations sum, product and a restricted star which is applied only to series for which the elements in the support all have the same connected alphabet.
Abstract: We will describe the recognizable formal power series over arbitrary semirings and in partially commuting variables, i.e. over trace monoids. We prove that the recognizable series are certain rational power series, which can be constructed from the polynomials by using the operations sum, product and a restricted star which is applied only to series for which the elements in the support all have the same connected alphabet. The converse is true if the underlying semi-ring is commutative. Moreover, if in addition the semiring is idempotent then the same result holds with a star restricted to series for which the elements in the support have connected (possibly different) alphabets. It is shown that these assumptions over the semiring are necessary. This provides a joint generalization of Kleene's. Schutzenberger's and Ochmanski's theorems.
8 citations
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TL;DR: In this article, the question of whether the sets of idempotents in weakly almost periodic compacti cations of (N, + +) and (Z, +) are closed is answered negatively.
Abstract: This paper answers negatively the question of whether the sets
of idempotents in the weakly almost periodic compacti�cations of (N; +) and
(Z; +) are closed.
8 citations
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TL;DR: In this paper, the authors studied the minimal size of a generating set of a finite algebra, which is called the growth rate of the algebra, and proved that a growth rate is either bounded by a polynomial in n, or exponential in n. They also gave a simple criterion for an algebra to have an exponential growth rate.
8 citations
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TL;DR: In this paper, the sub-power membership problem (SMP) was shown to be NP-complete for finite semigroups and the greatest variety of bands all of whose finite members induce a tractable SMP was determined.
8 citations