Hyperbolic Signatures in the Dynamics of Set-Theoretic Operations
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2025
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| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | This paper explores the geometric properties inherent in the algebraic structure of set operations, specifically complementation and intersection. We posit that the repeated application of these operations on subsets of a universal set generates a structure whose qualitative features are analogous to those found in hyperbolic geometry. By modeling the power set as a complemented distributive lattice (a Boolean algebra), we define a notion of distance based on the operational path length between elements. We demonstrate that sequences of intersections and complementations can lead to an exponential divergence of distinct elements, a hallmark of negative curvature. This divergence is analyzed by examining paths on the Hasse diagram of the Boolean lattice, which can be interpreted as a discrete space. The core thesis is that the process of logical refinement through intersection and categorical negation via complementation inherently creates a space of possibilities that expands hyperbolically. This perspective offers a novel geometric interpretation of logical operations and suggests that the abstract space of set-theoretic relations exhibits non-Euclidean characteristics. The implications of this model are discussed in the context of information theory, computational complexity, and the fundamental nature of logical spaces. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17469735 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Hyperbolic Signatures in the Dynamics of Set-Theoretic Operations SÉRGIO DE ANDRADE, PAULO Lattice Theory Boolean Algebra Set Theory Hyperbolic Geometry Logical Operations Discrete Geometry Set Complementation Set Intersection This paper explores the geometric properties inherent in the algebraic structure of set operations, specifically complementation and intersection. We posit that the repeated application of these operations on subsets of a universal set generates a structure whose qualitative features are analogous to those found in hyperbolic geometry. By modeling the power set as a complemented distributive lattice (a Boolean algebra), we define a notion of distance based on the operational path length between elements. We demonstrate that sequences of intersections and complementations can lead to an exponential divergence of distinct elements, a hallmark of negative curvature. This divergence is analyzed by examining paths on the Hasse diagram of the Boolean lattice, which can be interpreted as a discrete space. The core thesis is that the process of logical refinement through intersection and categorical negation via complementation inherently creates a space of possibilities that expands hyperbolically. This perspective offers a novel geometric interpretation of logical operations and suggests that the abstract space of set-theoretic relations exhibits non-Euclidean characteristics. The implications of this model are discussed in the context of information theory, computational complexity, and the fundamental nature of logical spaces. |
| title | Hyperbolic Signatures in the Dynamics of Set-Theoretic Operations |
| topic | Lattice Theory Boolean Algebra Set Theory Hyperbolic Geometry Logical Operations Discrete Geometry Set Complementation Set Intersection |
| url | https://doi.org/10.5281/zenodo.17469735 |