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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10 citations
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09 Jul 2009
TL;DR: In this article, the authors propose uniform idempotent right quasigroups and emergent algebras as an alternative to differentiable algesbras and prove a bijection between contractible groups and distributive uniform irqs (uniform quandles).
Abstract: Inspired from research subjects in sub-riemannian geometry and metric geometry, we propose uniform idempotent right quasigroups and emergent algebras as an alternative to differentiable algebras. Idempotent right quasigroups (irqs) are related with racks and quandles, which appear in knot theory (the axioms of a irq correspond to the first two Reidemeister moves). To any uniform idempotent right quasigroup can be associated an approximate differential calculus, with Pansu differential calculus in sub-riemannian geometry as an example. An emergent algebra A over a uniform idempotent right quasigroup X is a collection of operations such that each operation emerges from X, meaning that it can be realized as a combination of the operations of the uniform irq X, possibly by taking limits which are uniform with respect to a set of parameters. Two applications are considered: we prove a bijection between contractible groups and distributive uniform irqs (uniform quandles) and that some symmetric spaces in the sense of Loos may be seen as uniform quasigroups with a distributivity property.
10 citations
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TL;DR: It is shown that particular to power law input-to-state gain functions the deduction of the resulting sufficient condition for input- to-state stability may be performed efficiertly, using any suitable dynamic programming algorithm.
Abstract: In this paper a general nonlinear input-to-state stability small gain theory is described using idempotent analytic techniques. The theorem is proved within the context of the idempotent semiring K ⊂ End ○+ 0(R≥0), and may be regarded as an application of theoretical computer science techniques to systems and control theory. We show that particular to power law input-to-state gain functions the deduction of the resulting sufficient condition for input-to-state stability may be performed efficiertly, using any suitable dynamic programming algorithm. We indicate, through an example, how an analysis of the (weighted, directed) graph of the system complex gives a computable means to delimit (in an easily understood form) robust input-to-state stability bounds.
10 citations
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TL;DR: Iilton’s method is based on Evans’ Theorem that a finite partial latin square can be embedded in a latinsquare of order 2n, and gives an astonishingly simple construction which always embeds a partial idempotent latinSquare of order n in an idem Potent Latin square of order 4n.
10 citations
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TL;DR: In this article, a non-zero idempotent element can be represented as a sum of two nilpotent elements, where the latter is the sum of a ring and the former is a ring.
Abstract: Abstract In this paper we study in which rings a non-zero idempotent element can be presented as a sum of two nilpotent elements.
10 citations