SEARCH OF UNITS IN TENSORIAL ALGEBRA

D. I. Dzeboev

About the authors

D. I. Dzeboev

Дзебоев Д. И. – стажёр-исследователь, НИУ ВШЭ, Факультет компьютерных наук, Лаборатория моделирования и управления сложными системами, 109028, Москва, Покровский бульвар, 11. Область научных интересов: Тензориальная алгебра, гиперкомплексные числа, неассоциативные структуры, искусственный интеллект, нейронные сети. Orcid 0009-0008-4004-8750

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UDC: 512.554.1
Language: Russian

Abstract: The paper considers the analytical problem of finding unit elements in tensorial algebra — a generalized algebraic system of bilinear vector multiplication defined by an algebra tensor. It is shown that the action of any tensorial number in the algebra can be represented as a linear operator, and that the search for the unit element reduces to solving a system of linear equations where the contraction of the algebra tensor with an unknown vector yields the identity matrix. General formulas for the left and right units are derived in terms of Moore –Penrose pseudoinverses. It is proven that if at least one right and one left unit exist, they coincide, forming a unique two -sided (commutative) unit that renders the algebra unital.

Keywords: tensorial algebra, algebra tensor, algebraic unit, right and left unit, Moore– Penrose pseudoinverse, linear operator, unital structure

References

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Issue: 4(39)'2025
Section: ARTIFICIAL INTELLIGENCE AND MACHINE LEARNING
How to cite: D. I. Dzeboev. SEARCH OF UNITS IN TENSORIAL ALGEBRA // Problems of Artificial Intelligence. - 2025. - № 4 (39). - P. 27-35. - https://paijournal.guiaidn.ru/en/2025/4(39)-3.html

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