A Theory of Congruences and Birkhoff’s Theorem for Matroids

A congruence is defined for a matroid. This leads to suitable versions of the algebraic isomorphism theorems for matroids. As an application of the congruence theory for matroids, a version of Birkhoff’s Theorem for matroids is given which shows that every nontrivial matroid is a subdirect product of subdirectly irreducible matroids.

Bott–Thom isomorphism, Hopf bundles and Morse theory

Based on Morse theory for the energy functional on path spaces we develop a deformation theory for mapping spaces of spheres into orthogonal groups. This is used to show that these mapping spaces are weakly homotopy equivalent, in a stable range, to mapping spaces associated to orthogonal Clifford representations. Given an oriented Euclidean bundle $$V \rightarrow X$$ V → X of rank divisible by four over a finite complex X we derive a stable decomposition result for vector bundles over the sphere bundle $$\mathord {{\mathbb {S}}}( \mathord {{\mathbb {R}}}\oplus V)$$ S ( R ⊕ V ) in terms of vector bundles and Clifford module bundles over X. After passing to topological K-theory these results imply classical Bott–Thom isomorphism theorems.

Filters of strong Sheffer stroke non-associative MV-algebras

In this paper, at first we study strong Sheffer stroke NMV-algebra. For getting more results and some classification, the notions of filters and subalgebras are introduced and studied. Finally, by a congruence relation, we construct a quotient strong Sheffer stroke NMV-algebra and isomorphism theorems are proved.

Suatu Kajian Tentang B-Aljabar

Abstrak. Penelitian ini merupakan penelitian kajian kepustakaan yang bertujuan untuk mengkaji konsep dan sifat-sifat terkait B-Aljabar. Konsep B-Aljabar dalam penelitian ini berdasarkan penelitian yang telah dilakukan oleh Neggers dan Kim serta Allen. Seluruh pembahasan dalam penelitian ini menggunakan himpunan tegas, baik himpunan berhingga maupun himpunan tidak berhingga. Hasilnya, dapat diberikan bukti yang lebih lengkap dari sifat-sifat B-Aljabar serta hubungannya dengan grup. Suatu grup dengan definisi operasi khusus dan elemen identitas merupakan B-Aljabar. Lebih lanjut dapat diturunkan beberapa teorema grup kedalam B-Aljabar seperti pemetaan natural dan Teorema Isomorfisma 1 yang dalam pembuktiannya memiliki kemiripan dengan pembuktian pada grup dengan tetap menggunakan sifat-sifat B-Aljabar itu sendiri.Kata Kunci: B-Aljabar, B-Subaljabar, B-Homomorfisma, B-IsomorfismaAbstract. This research is a literature studies that aims at reviewing the concepts and properties of B-Algebras. The concept of B-Algebras in this article is based on research that has been done by Neggers and Kim and Allen. All discussions in this article use the firm sets, both finite sets and infinite sets. As a result, more complete evidence of the properties of B-Algebras can be given and its relationship with the group. A group with a specific operation and has as an identity element is a B-Algebras. Moreover, a number of group theorems can be derived into B-Algebra such as natural mapping and the First Isomorphism Theorems which in their proof have similarities to the proofs of groups while still using the properties of B-Algebra itself.Keywords: B-Algebras, B-Subalgebras, B-Homomorphism, B-Isomorphism

Fuzzy mixed graphs and its application to identification of COVID19 affected central regions in India

In the recent phenomenon of social networks, both online and offline, two nodes may be connected, but they may not follow each other. Thus there are two separate links to be given to capture the notion. Directed links are given if the nodes follow each other, and undirected links represent the regular connections (without following). Thus, this network may have both types of relationships/ links simultaneously. This type of network can be represented by mixed graphs. But, uncertainties in following and connectedness exist in complex systems. To capture the uncertainties, fuzzy mixed graphs are introduced in this article. Some operations, completeness, and regularity and few other properties of fuzzy mixed graphs are explained. Representation of fuzzy mixed graphs as matrix and isomorphism theorems on fuzzy mixed graphs are developed. A network of COVID19 affected areas in India are assumed, and central regions are identified as per the proposed theory.