Remarks on some Limit Geometric Properties related to an Idempotent and Non-Associative Algebraic Structure
Walter Briec
TL;DR
This work develops a comprehensive limit-based framework for an idempotent, non-associative algebraic structure that extends Max-Times, using $p\to\infty$ dequantization to define a consistent vector-space, inner product, and determinant in the limit. It introduces an ultrametric distance $d_{\boxplus}$ and a rich geometry of ultrametric balls, together with a generalized notion of orthogonality and a Pythagorean-type relation in the limit. The paper further elaborates limit-based trigonometric notions, including pseudo-cosine and pseudo-sine, and constructs a parametrization of the unit square, with a complex-field extension and a limiting product $\boxtimes$. Finally, it provides an algebraic description of the Kuratowski–Painlevé limit of lines, including half-line decompositions and a parallel-lines framework, revealing unusual geometric phenomena such as infinite intersections of parallel lines and multiple lines through two points. These results connect idempotent convexity, ultrametric geometry, and non-associative line theory to Max-Times/Max-Plus systems and their determinant-like limits, with implications for complex plane geometry under dequantization.
Abstract
This article analyzes the geometric properties of an idempotent, non-associative algebraic structure that extends the Max-Times semiring. This algebraic structure is useful for studying systems of Max-Times and Max-Plus equations, employing an appropriate notion of a non-associative determinant. We consider a connected ultrametric distance and demonstrate that it implies, among other properties, an analogue of the Pythagorean relation. To this end, we introduce a suitable notion of a right angle between two vectors and investigate a trigonometric concept associated with the Chebyshev unit ball. Following this approach, we explore the potential implications of these properties in the complex plane. We provide an algebraic definition of a line passing through two points, which corresponds to the Painlevé-Peano-Kuratowski limit of a sequence of generalized lines. We establish that this definition leads to distinctive geometric properties; in particular, two distinct parallel lines may share an infinite number of points.