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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, the integrability of quadrational Yang-Baxter maps and known integrable multi-quadratic quad equations are unified by combining theory from these two classes of quad-graph models, and a natural extension of the associated lattice geometry is obtained.
Abstract: A transformation is obtained which completes the unification of quadrirational Yang-Baxter maps and known integrable multi-quadratic quad equations. By combining theory from these two classes of quad-graph models we find an extension of the known integrability feature, and show how this leads subsequently to a natural extension of the associated lattice geometry. The extended lattice is encoded in a birational representation of a particular sequence of Coxeter groups. In this setting the usual quad-graph is associated with a subgroup of type BC_n, and is part of a larger and more symmetric ambient space. The model also defines, for instance, integrable dynamics on a triangle-graph associated with a subgroup of type A_n, as well as finite degree-of-freedom dynamics, in the simplest cases associated with affine-E6 and affine-E8 subgroups. Underlying this structure is a class of biquadratic polynomials, that we call idempotent, which express the trisection of elliptic function periods algebraically via the addition law.

16 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that the number of cases in which a linear combination of idempotent matrices P 1, P 2 with nonzero complex numbers c 1, c 2 is the group involutory matrix is infinite.
Abstract: Coll and Thome [Coll, C. and Thome, N., 2003, Oblique projectors and group involutory matrices. Applied Mathematics and Computation, 140, 517–522] considered the problem of ‘when a linear combination of nonzero different complex idempotent matrices P 1, P 2, with nonzero complex numbers c 1, c 2, is the group involutory matrix?’ According to the solution provided therein as Theorem 1, it is possible in a finite number of cases, each characterized by definite values of scalars c 1 and c 2. In the present article, this problem is revisited and it is shown that the actual number of cases, in which a linear combination of interest is the group involutory matrix, is infinite and that there is certain freedom regarding values of c 1 and c 2.

16 citations

Journal ArticleDOI
TL;DR: For a suitable series of idempotent ideals, a method of constructing tilting modules of finite projective dimension is given in this article, where the tilting module is constructed for a suitable set of ideals.

16 citations

Book ChapterDOI
TL;DR: A very brief introduction to mathematics of semirings (including idempotent and tropical mathematics) is presented and concrete applications to optimization problems, idem Potent linear algebra and interval analysis are indicated.
Abstract: This isaut]Grigory L. Litvinovaut]Victor P. Maslovaut]Anatoly Ya. Rodionovaut]Andrei N. Sobolevskii a survey paper on applications of mathematics of semirings to numerical analysis and computing. Concepts of universal algorithm and generic program are discussed. Relations between these concepts and mathematics of semirings are examined. A very brief introduction to mathematics of semirings (including idempotent and tropical mathematics) is presented. Concrete applications to optimization problems, idempotent linear algebra and interval analysis are indicated. It is known that some nonlinear problems (and especially optimization problems) become linear over appropriate semirings with idempotent addition (the so-called idempotent superposition principle). This linearity over semirings is convenient for parallel computations.

16 citations

Journal ArticleDOI
TL;DR: In this paper, the authors investigate the structure of the twisted Brauer monoid and give necessary and sufficient conditions for an ideal to be idempotent generated, and obtain formulae for the rank (smallest size of a generating set) of each principal ideal.
Abstract: We investigate the structure of the twisted Brauer monoid , comparing and contrasting it with the structure of the (untwisted) Brauer monoid . We characterize Green's relations and pre-orders on , describe the lattice of ideals and give necessary and sufficient conditions for an ideal to be idempotent generated. We obtain formulae for the rank (smallest size of a generating set) and (where applicable) the idempotent rank (smallest size of an idempotent generating set) of each principal ideal; in particular, when an ideal is idempotent generated, its rank and idempotent rank are equal. As an application of our results, we describe the idempotent generated subsemigroup of (which is not an ideal), as well as the singular ideal of (which is neither principal nor idempotent generated), and we deduce that the singular part of the Brauer monoid is idempotent generated, a result previously proved by Maltcev and Mazorchuk.

16 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023106
2022263
202184
2020100
201991
201892