Strong regularity of matrices — a survey of results
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paper_title: A condition for the strong regularity of matrices in the minimax algebra

paper_content:

Abstract Columns of a matrix A in the minimax algebra are called strongly linearly independent if for some b the system of equations A⊗x = b is uniquely solvable (cf. [3]). This paper presents a condition which is necessary and sufficient for the strong linear independence of columns of a given matrix in the minimax algebra based on a dense linearly ordered commutative group. In the case of square matrices an O(n3) method for checking this property as well as for finding at least one b such that A⊗x = b is uniquely solvable is derived. A connection with the classical assignment problem is formulated.
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paper_title: A condition for the strong regularity of matrices in the minimax algebra

paper_content:

Abstract Columns of a matrix A in the minimax algebra are called strongly linearly independent if for some b the system of equations A⊗x = b is uniquely solvable (cf. [3]). This paper presents a condition which is necessary and sufficient for the strong linear independence of columns of a given matrix in the minimax algebra based on a dense linearly ordered commutative group. In the case of square matrices an O(n3) method for checking this property as well as for finding at least one b such that A⊗x = b is uniquely solvable is derived. A connection with the classical assignment problem is formulated.
---
paper_title: A condition for the strong regularity of matrices in the minimax algebra

paper_content:

Abstract Columns of a matrix A in the minimax algebra are called strongly linearly independent if for some b the system of equations A⊗x = b is uniquely solvable (cf. [3]). This paper presents a condition which is necessary and sufficient for the strong linear independence of columns of a given matrix in the minimax algebra based on a dense linearly ordered commutative group. In the case of square matrices an O(n3) method for checking this property as well as for finding at least one b such that A⊗x = b is uniquely solvable is derived. A connection with the classical assignment problem is formulated.
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Title: Strong Regularity of Matrices — A Survey of Results

Section 1: Introduction
Description 1: Provide an introduction to the problem area, including motivation, key concepts, and an illustrative example.

Section 2: Max-algebra
Description 2: Discuss the foundational concepts of max-algebra including definitions, properties, and basic operations.

Section 3: Linear Equations and the Eigenproblem in Max-algebra
Description 3: Detailed exploration of the linear equations and eigenvalue problems within the max-algebra framework.

Section 4: Linear Independence, Rank, and Regularity in Max-algebra
Description 4: Define and examine linear independence, rank, and regularity of matrices in the context of max-algebra.

Section 5: Strong Regularity of Matrices in Max-algebra
Description 5: Summarize the results related to checking the strong regularity of matrices, including methodological and algorithmic approaches.

Section 6: Strong Regularity in Bottleneck Algebra
Description 6: Investigate the strong regularity in a bottleneck algebra setting, including necessary and sufficient conditions, algorithms, and computational complexity.

Section 7: Conclusion and Open Questions
Description 7: Summarize the main findings of the survey and outline open questions and directions for future research.