Determinants and limit systems in some idempotent and non-associative algebraic structure

Abstract

This paper considers an idempotent and symmetrical algebraic structure as well as some closely related concepts. 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. 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 whose zeros are related to the eigenvalues of a matrix with nonnegative entries.