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[3] viXra:2403.0051 [pdf] submitted on 2024-03-12 23:57:58
Authors: Shao-Dan Lee
Comments: 6 Pages.
We construct an algebra A such that A has a nonempty finite set Δ of associative and commutative binary operations. Then we may define an ideal with respect to a nonempty subset of Δ. If some hypotheses are satisfied, then we have that a union of the ideals is an ideal. An ideal M is maximal with respect to a subset of Δ if there is not an ideal J ≠ A such that J contains M. And an algebra is local with respect to a subset of Δ if it has a unique maximal ideal. Suppose that the algebra A is local with respect to Φ and Ψ, M and N are the maximal ideals, respectively, and J is an ideal with respect to Φ∪Ψ. Then we have that J ⊆ M∩ N if some conditions hold. Let A be a local algebra with respect to Φ, M the maximal ideal. For all Ψ with Φ ⊆ Ψ⊂ Δ, if M is an ideal with respect to Ψ, then A is local with respect to Ψ. A preimage of an ideal with respect to Φ under a homomorphism is an ideal with respect to Φ.
Category: Algebra
[2] viXra:2403.0014 [pdf] submitted on 2024-03-05 15:38:35
Authors: Robert Benjamin Easter, Daranee Pimchangthong
Comments: 23 Pages.
In Geometric Algebra, the degenerate-metric algebra G(3,0,1) is known as the Projective Geometric Algebra (PGA) for 3D space (3DPGA). In PGA, there is a point-based geometric algebra (point-based PGA) and a plane-based geometric algebra (plane-based PGA). Both algebras have homogeneous geometric entities for points, lines, and planes. The two algebras of PGA are dual to each other through a new geometric entity dualization operation J_e, which is introduced in this paper as its main subject and contribution. The new dualization J_e is an anti-involution with inverse −J_e = D_e. Using J_e, the dual of a point-based PGA entity is its corresponding plane-based PGA entity representing the same geometry (point, line, or plane) with the same orientation. Using D_e = −J_e, the inverse dual (undual) of a plane-based PGA entity is its corresponding point-based PGA entity with the same orientation. The new dualization operation Je maintains the correct orientation of an entity. J_e is defined by a table of duals that are found empirically by observation to maintain correct entity orientation through the dualization. We define a Hodge star dualization operation to be purely an involution, or else purely an anti-involution, between all basis blades and their dual basis blades. As an anti-involution, J_e is also implemented by algebraic methods using Hodge star dualizations in non-degenerate algebras that correspond to PGA. In the prior literature, there are other definitions for the duals in PGA that may not maintain the correct entity orientation and are different than J_e.
Category: Algebra
[1] viXra:2403.0012 [pdf] submitted on 2024-03-05 15:50:27
Authors: Robert Benjamin Easter, Daranee Pimchangthong
Comments: 28 Pages.
In Geometric Algebra, the algebra G(3,0,1) is known as PGA, the plane-based and point-based geometric algebras, or projective geometric algebra, of points, lines, and planes in 3D space. The even-grades subalgebra of PGA, which we call Dual Quaternion Geometric Algebra (DQGA), represents Dual Quaternion Algebra (DQA). In the plane-based algebra of PGA, there are entities for points, lines, and planes and many operations on them, including dualization to the point-based entities, reflections in planes, rotations, translations, projections, rejections, and intersections (meet products). In this paper, we derive a complete set of identities that relate all of the plane-based entities and operations in PGA to their corresponding entities and operations in DQGA. Therefore, this paper contributes into the literature on dual quaternions and PGA the complete details on how to use DQA or DQGA as a geometric algebra of points, lines, and planes with many useful operations. All DQGA entities and operations are defined or derived such that the orientations of the entities are maintained correctly through all of the operations. We also define three new part operators for taking the point, line, or plane part of a dual quaternion, which may improve the computational efficiency of intersection (meet) operations. Dual quaternions already have some applications in computer graphics and kinematics. This paper expands on the understanding of dual quaternions and introduces DQA as a versatile geometric algebra of points, lines, and planes with many new operations that do not appear in prior literature, expanding the possible applications of dual quaternions.
Category: Algebra
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