Determinants and Limit Systems in some Idempotent and Non-Associative Algebraic Structure

TL;DR

This paper explores determinants, limit systems, and eigenvalues in idempotent and non-associative algebraic structures, extending classical results and introducing new algebraic concepts and formulas.

Contribution

It introduces a new notion of determinant, derives a Cramer formula for limit systems, and develops a method to find eigenvalues in nonnegative matrices within these algebraic frameworks.

Findings

Derived a Cramer formula for limit systems from Hadamard products

Introduced a new determinant concept for idempotent algebraic structures

Constructed a semi-continuous polynomial to compute eigenvalues

Abstract

This paper considers an idempotent and symmetrical algebraic structure as well as some closely related concept. A special notion of determinant is introduced and a Cramer formula is derived for a class of limit systems derived from the Hadamard matrix product and we give the algebraic form of a sequence of hyperplanes passing through a finite number of points. Thereby, some standard results arising for Max-Times systems with nonnegative entries appear as a special case. The case of two sided systems is also analyzed. In addition, a notion of eigenvalue in limit is considered. It is shown that one can construct a special semi-continuous regularized polynomial to find the eigenvalues of a matrix with nonnegative entries.