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 paper, an axiomatic characterization of idempotent discrete uninorms by means of three conditions only: conservativeness, symmetry, and non-decreasing monotonicity is provided.
30 citations
TL;DR: In this article, it was shown that for every n × n matrix A over the field Z 2 there exists an idempotent matrix E such that (A − E ) 4 = 0.
30 citations
TL;DR: A stronger version of the IFS approach to arbitrary dimensions d≥2 is presented and it is shown that for every s∈(1,d) the authors can find a d-dimensional copula whose support has Hausdorff dimension s.
30 citations
TL;DR: In this paper, the spectrum of idempotent cyclic projectors is characterized in terms of a suitable extension of Hilbert's projective metric, and the authors deduce as a corollary of their main results the cyclic analogue of Helly's theorem.
Abstract: Semimodules over idempotent semirings like the max-plus or tropical semiring have much in common with convex cones. This analogy is particularly apparent in the case of subsemimodules of the n-fold cartesian product of the max-plus semiring it is known that one can separate a vector from a closed subsemimodule that does not contain it. We establish here a more general separation theorem, which applies to any finite collection of closed semimodules with a trivial intersection. In order to prove this theorem, we investigate the spectral properties of certain nonlinear operators called here idempotent cyclic projectors. These are idempotent analogues of the cyclic nearest-point projections known in convex analysis. The spectrum of idempotent cyclic projectors is characterized in terms of a suitable extension of Hilbert's projective metric. We deduce as a corollary of our main results the idempotent analogue of Helly's theorem.
29 citations
TL;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: Kleene's theorem on the coincidence of regular and rational languages in free monoids has been generalized by Schutzenberger to a description of the recognizable formal power series in noncommuting variables over arbitrary semirings and by Ochmanski to a characterization of the recognizable languages in trace monoids. 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 semiring 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.
29 citations