Algebra

2206 Submissions

[4] viXra:2206.0105 [pdf] submitted on 2022-06-20 23:28:56

Some Facts about Relations and Operations of Algebras

Authors: Shao-Dan Lee
Comments: 5 Pages.

Let A be a σ-algebra. Suppose that Θ is a congruence of A. Then Θ is a subalgebra of A×A. If φ is an automorphism from A to A, then (φ,φ) is an automorphism of A×A. And it is obvious that (φ,φ)(Θ) is a congruence of A. Let B be a σ-algebra and ψ a homomorphism from A to B. Then B′ := ψ(A) is a subalgebra of B. And (ψ,ψ)(Θ) is a congruence of B′. If ψ is an epimorphism, then (ψ,ψ)(Θ) is a congruence of B. Suppose that A is a category of all σ-algebras. Let A,B ∈ A and ψ: A → B be a homomorphism. Then the pullback A ⊓B A is isomorphic to a congruence of A. An n-ary relation of an algebra A is a subset of An. If satisfies some conditions, then is a subalgebra of An. The set of languages is a lattice. If is the set of the compositions of the operations in a language σ, then is an algebra.
Category: Algebra

[3] viXra:2206.0099 [pdf] submitted on 2022-06-19 22:14:58

Larger Types of Infinities and Its Impact on Society

Authors: Murilo Leandro
Comments: 3 Pages.

In this paper, we consider an extended real number denoting "larger types of infinity" through elements of a polynomial in a way that justify concepts as infinity plus one, or multiplication of infinities. Then, we equip the set with sums and multiplication such that it forms a ring, and finally, define its order relations.
Category: Algebra

[2] viXra:2206.0072 [pdf] submitted on 2022-06-14 06:15:00

Special Affine Fourier Transform for Space-Time Algebra Signals in Detail

Authors: Eckhard Hitzer
Comments: 18 Pages. To be published in Adv. Appl. Clifford Algebras, 2022.

We generalize the space-time Fourier transform (SFT) [1] to a special affine Fourier transform (SASFT, also known as offset linear canonical transform) for 16-dimensional space-time multivector Cl(3,1)-valued signals over the domain of space-time (Minkowski space) R^{3,1}. We establish how it can be computed in terms of the SFT, and introduce its properties of multivector coefficient linearity, shift and modulation, inversion, Rayleigh (Parseval) energy theorem, partial derivative identities, a directional uncertainty principle and its specialization to coordinates. All important results are proven in full detail. [1] E. Hitzer, Quaternion Fourier Transform on Quaternion Fields and Generalizations. Adv. Appl. Clifford Algebras 17(3), pp. 497-517 (2007), DOI: https://doi.org/10.1007/s00006-007-0037-8.
Category: Algebra

[1] viXra:2206.0040 [pdf] replaced on 2025-09-19 14:57:52

Division by Zero

Authors: Tim Olsson
Comments: 1 Page. Sorry for budding the error so late but courtesy of Bush et al. Please delete the "_ljtL" part in the .pdf name. I don't know why it's there...

An algebraic theory of extended complex numbers.
Category: Algebra