Ultra Exponent Matrices

• Date: 2026
• Main Author: Зеленський, Олексій
• Format: Article
• Language: English
• Published: Кам'янець-Подільський національний університет імені Івана Огієнка 2026
• Online Access: https://mcm-math.kpnu.edu.ua/article/view/361006
• Journal Title: Mathematical and computer modelling. Series: Physical and mathematical sciences

The paper studies ultra exponent matrices, that is, reduced exponent matrices for which the usual triangle inequality is strengthened to an ultrametric-type inequality. It is proved that every ultra exponent matrix is an exponent matrix, and that every exponent matrix whose entries belong to {0, 1} is an ultra exponent matrix. The behaviour of ultra matrices under elementary transformations is analysed: transformations of the first type do not preserve the ultra property in general, while simultaneous permutations of rows and columns do preserve it. It is shown that a quiver obtained from a reduced ultra exponent matrix with at least one entry greater than one is not rigid. A counterexample demonstrates that not every admissible quiver with a loop at every vertex can be represented by an ultra exponent matrix. Several structural characterizations are also established, including monotone deformations preserving the associated quiver, a filtration description in terms of transitive threshold relations, a connection between 0-1 ultra matrices and partial orders, a minimax interpretation, and closure under componentwise maximum. From the viewpoint of mathematical modelling, ultra exponent matrices may be interpreted as discrete directed distance models with bottleneck-type constraints, where the value assigned to a transition is determined by the strongest restriction along admissible paths rather than by an additive cost. This makes them useful for modelling hierarchical systems, priority relations, minimax optimization, constrained network flows, clustering-like structures, and algebraic or combinatorial systems whose essential information is encoded by admissible weighted quivers. The obtained results provide tools for comparing such models and for reducing redundant representations without losing the underlying directed structure.