Skew-Field of Trace-Preserving Endomorphisms, of Translation Group in Affine Plane

Abstract

In this paper we will show how to constructed an Skew-Field with trace-preserving endomorphisms of the affine plane. Earlier in my paper, we doing a detailed description of endomorphisms algebra and trace-preserving endomorphisms algebra in an affine plane, and we have constructed an associative unitary ring for which trace-preserving endomorphisms. In this paper we formulate and prove an important Lemma, which enables us to construct a particular trace-preserving endomorphism, with the help of which we can construct the inverse trace-preserving endomorphisms of every trace-preserving endomorphism. At the end of this paper we have proven that the set of trace-preserving endomorphisms together with the actions of 'addition' and 'composition' (which is in the role of 'multiplication') forms a skew-field.

arXiv comment11 pages. arXiv admin note: substantial text overlap with arXiv:2003.09528

References in corpus (5)

• The Transform of a line of Desargues Affine Plane in an additive Group of its Points
• Isomorphic-Dilations of the skew-fields constructed over parallel lines in the Desargues affine plane
• The Endomorphisms Algebra of Translations Group and Associative Unitary Ring of Trace-Preserving Endomorphisms in Affine Plane
• Ordered Line and Skew-Fields in the Desargues Affine Plane
• A constructive approach to affine and projective planes

Cited by in corpus (1)

• The Determinant of Cubic-Matrix of order 2 and order 3: Some basic Properties and Algorithms