The Lattice of Involutions: A Geometric Classification of Complementary Symmetry Structures
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| Format: | Recurso digital |
| Sprache: | Englisch |
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2026
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| _version_ | 1866902031625617408 |
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| author | Gonzalez-Granda Fernandez, Eduardo |
| author_facet | Gonzalez-Granda Fernandez, Eduardo |
| contents | <p>This paper introduces a geometric classification of complementary symmetry<br>structures through the lattice generated by distinguished involutions. Building upon<br>the framework of complementary symmetry groups in intuitionistic fuzzy dynamical<br>systems, we define L(G, I) as the lattice generated by the set of distinguished involutions<br>I of a complementary symmetry group G, ordered by subgroup inclusion.<br>This lattice provides a structural invariant finer than group isomorphism, capturing<br>the internal geometry of complementary oppositions. For commutative involutiongenerated<br>groups, we prove L(G, I) ∼= P(I), establishing a complete Boolean classification<br>where binary, quaternary, octonary, and hexadecimary structures correspond<br>to segments, squares, cubes, and hypercubes respectively. We further introduce a<br>metric on complementary structures via the Gromov-Hausdorff distance between<br>their involution lattices, enabling quantification of structural approximation and deformation.<br>This geometric approach provides a rigorous foundation for classifying,<br>comparing, and evolving complementary structures across mathematical and applied<br>domains.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18489029 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Lattice of Involutions: A Geometric Classification of Complementary Symmetry Structures Gonzalez-Granda Fernandez, Eduardo complementary symmetry involution lattice Boolean classification structural metric hierarchical organization hypercube geometry <p>This paper introduces a geometric classification of complementary symmetry<br>structures through the lattice generated by distinguished involutions. Building upon<br>the framework of complementary symmetry groups in intuitionistic fuzzy dynamical<br>systems, we define L(G, I) as the lattice generated by the set of distinguished involutions<br>I of a complementary symmetry group G, ordered by subgroup inclusion.<br>This lattice provides a structural invariant finer than group isomorphism, capturing<br>the internal geometry of complementary oppositions. For commutative involutiongenerated<br>groups, we prove L(G, I) ∼= P(I), establishing a complete Boolean classification<br>where binary, quaternary, octonary, and hexadecimary structures correspond<br>to segments, squares, cubes, and hypercubes respectively. We further introduce a<br>metric on complementary structures via the Gromov-Hausdorff distance between<br>their involution lattices, enabling quantification of structural approximation and deformation.<br>This geometric approach provides a rigorous foundation for classifying,<br>comparing, and evolving complementary structures across mathematical and applied<br>domains.</p> |
| title | The Lattice of Involutions: A Geometric Classification of Complementary Symmetry Structures |
| topic | complementary symmetry involution lattice Boolean classification structural metric hierarchical organization hypercube geometry |
| url | https://doi.org/10.5281/zenodo.18489029 |