International Journal of Topology

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Int. J. Topol., Volume 2, Issue 3 (September 2025) – 2 articles

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28 pages, 338 KiB

Open AccessArticle

Superhypermagma, Lie Superhypergroup, Quotient Superhypergroups, and Reduced Superhypergroups

by Takaaki Fujita

Int. J. Topol. 2025, 2(3), 10; https://doi.org/10.3390/ijt2030010 - 8 Jul 2025

Viewed by 148

Abstract

Classical algebraic structures—such as magmas, groups, and Lie groups—are characterized by increasingly strong requirements in binary operation, ranging from no additional constraints to associativity, identity, inverses, and smooth-manifold structures. The hyperstructure paradigm extends these notions by allowing the operation to return subsets of elements, giving rise to hypermagmas, hypergroups, and Lie hypergroups, along with their variants such as quotient, reduced, and fuzzy hypergroups. In this work, we introduce the concept of superhyperstructures, obtained by iterating the powerset construction, and develop the theory of superhypermagmas and Lie superhypergroups. We further define and analyze quotient superhypergroups, reduced superhypergroups, and fuzzy superhypergroups, exploring their algebraic properties and interrelationships. Full article

(This article belongs to the Special Issue Feature Papers in Topology and Its Applications)

48 pages, 944 KiB

Open AccessArticle

Spaces of Polynomials as Grassmanians for Immersions and Embeddings

by Gabriel Katz

Int. J. Topol. 2025, 2(3), 9; https://doi.org/10.3390/ijt2030009 - 24 Jun 2025

Viewed by 153

Abstract

Let Y be a smooth compact n-manifold. We studied smooth embeddings and immersions

β : M \to R \times Y

of compact n-manifolds M such that

β (M)

avoids some priory chosen closed poset

Θ

of tangent patterns to [...] Read more.

Let Y be a smooth compact n-manifold. We studied smooth embeddings and immersions

β : M \to R \times Y

of compact n-manifolds M such that

β (M)

avoids some priory chosen closed poset

Θ

of tangent patterns to the fibers of the obvious projection

π : R \times Y \to Y

. Then, for a fixed Y, we introduced an equivalence relation between such

β

’s; creating a crossover between pseudo-isotopies and bordisms. We called this relation quasitopy. In the presented study of quasitopies, the spaces

P_{d}^{c Θ}

of real univariate polynomials of degree d with real divisors, whose combinatorial patterns avoid a given closed poset

Θ

, play the classical role of Grassmanians. We computed the quasitopy classes

Q_{d}^{emb} (Y, c Θ)

Θ

-constrained embeddings

β

in terms of homotopy/homology theory of spaces Y and

P_{d}^{c Θ}

. We proved also that the quasitopies of embeddings stabilize, as

d \to \infty

. Full article

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Int. J. Topol., Volume 2, Issue 3 (September 2025) – 2 articles

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