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.

A complete closed-form solution to a tropical extremal problem

Abstract: A multidimensional extremal problem in the idempotent algebra setting is considered which consists in minimizing a nonlinear functional defined on a finite-dimensional semimodule over an idempotent semifield. The problem integrates two other known problems by combining their objective functions into one general function and includes these problems as particular cases. A new solution approach is proposed based on the analysis of linear inequalities and spectral properties of matrices. The approach offers a comprehensive solution to the problem in a closed form that involves performing simple matrix and vector operations in terms of idempotent algebra and provides a basis for the development of efficient computational algorithms and their software implementation.

Idempotent interval analysis and optimization problems

Abstract: Many problems in optimization theory are strongly nonlinear in the traditional sense but possess a hidden linear structure over suitable idempotent semirings. After an overview of `Idempotent Mathematics' with an emphasis on matrix theory, interval analysis over idempotent semirings is developed. The theory is applied to construction of exact interval solutions to the interval discrete stationary Bellman equation. Solution of an interval system is typically NP-hard in the traditional interval linear algebra; in the idempotent case it is polynomial. A generalization to the case of positive semirings is outlined.

On the dimension of polynomial semirings

Abstract: In our previous work, motivated by the study of tropical polynomials, a definition for prime congruences was given for an arbitrary commutative semiring. It was shown that for additively idempotent semirings this class exhibits some analogous properties to prime ideals in ring theory. The current paper focuses on the resulting notion of Krull dimension, which is defined as the length of the longest chain of prime congruences. Our main result states that for any additively idempotent semiring $A$, the semiring of polynomials $A[x]$ and the semiring of Laurent polynomials $A(x)$, we have $\dim A[x] = \dim A(x) = \dim A + 1$.

A method for the construction of orthogonal spin eigenfunctions

Abstract: A method for the construction of the essentially idempotent and Hermitian diagonal elements of the matric algebra of the permutation group Sn is proposed. For the irreducible representation [λ] = [λ1, λ2] characterising a spin state S of an n-electron system, it is found that this method generates the complete set of spin projections from the appropriate primitive spin functions. The method is applied to a 7-electron system in the spin state S = MS = 1/2 and the results are listed in the Appendix.

Lattice Ordered Polynomial Algebras

Abstract: In this paper we define a lattice order on a set F of binary functions. We then provide necessary and sufficient conditions for the resulting algebra \(\mathfrak{L}\)F to be a distributive lattice or a Boolean algebra. We also prove a ‘Cayley theorem’ for distributive lattices by showing that for every distributive lattice \(\mathfrak{L}\), there is an algebra \(\mathfrak{L}\)F of binary functions, such that \(\mathfrak{L}\) is isomorphic to\(\mathfrak{L}\)F and we show that \(\mathfrak{L}\)F is a distributive lattice iff the operations ∨ and ∧ are idempotent and cummutative, showing that this result cannot be generalized to non-distributive lattices or quasilattices without changing the definitions of ∨ and ∧. We also examine the equational properties of an Algebra \(\mathfrak{U}\) for which \(\mathfrak{L}_\mathfrak{U}\), now defined on the set of binary \(\mathfrak{U}\)-polynomials is a lattice or Boolean algebra.