Linear algebra over T-pairs

TL;DR

This paper develops a linear algebra framework over semiring pairs, addressing rank equality issues and applications to tropical algebra, hyperrings, and fuzzy rings, with new counterexamples and conditions for rank equality.

Contribution

It introduces a general algebraic framework for semiring pairs, explores rank properties, and provides new counterexamples and conditions for matrix rank equality in this setting.

Findings

Counterexample to rank equality in pairs of the second kind.

Conditions under which row, column, and submatrix ranks are equal.

Extensions of classical linear algebra results like Cramer's rule to semiring pairs.

Abstract

This is part of an ongoing project to find a general algebraic framework for semiring theory. The structure theory of semirings is quite challenging, largely because of the lack of negation, and such basic properties such as unique factorization of polynomials, multiplicativity of determinants, and the characteristic polynomial of a matrix, all fail. (In fact in the max-plus algebra, the sum of two nonzero elements is never zero!) Consequently 0 is replaced by a distinguished T-submodule $ A_0$ of $ A,$ and $( A, A_0)$ is called a ``pair.'' This paper treats linear algebra over a (not necessarily distributive) semiring pair, with a range of applications to tropical algebra as well as related areas such as hyperrings and fuzzy rings. We turn to matrices and the question of whether the row rank, column rank, and submatrix rank of a matrix are equal. The submatrix rank is less than or equal to the row rank and the column rank in many cases, including ``metatangible pairs'' with unique negation, but there is a counterexample to equality, discovered some time ago by the second author, which we provide in a more general setting (``pairs of the second kind'') that includes the hyperfield of signs. We do find situations when equality holds, encompassing results by Akian, Gaubert, Guterman, Izhakian, Knebusch, and Rowen, including versions of Cramer's rule. We pay special attention to the question of Baker and Zhang whether $n+1$ vectors of length $n$ need be dependent. At the conclusion of the main part, we consider surpassing relations, which permit us to tighten our results. The categorical setting is given in the appendix.