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DOI: 10.46698/i3178-1119-0009-t
One-Sided Dual Schemes
Shishkin, A. B.
Vladikavkaz Mathematical Journal 2020. Vol. 22. Issue 3.
Abstract: The phenomenon of duality appears in all areas of mathematics and is closely related to the phenomenon of equivalence. These phenomena complement each other and are used to transfer various mathematical statements from one area of mathematics to another and vice versa (dual and equivalent transitions). The main difference between duality and equivalence is the use of involution. An involution of an object is a transformation of an object whose action is eliminated by an inverse transformation, that is, an inverse transformation restores an object. Any involution generates its duality, which is affirmed by the corresponding duality theorem. Duality theorems are two-sided. They allow for dual transitions to one and the other. Weaken the conditions for involution and assume that its repeated action restores the object only by half (instead of equality, we obtain inequality). In this case, for such a complete restoration of an object, two such involutions are required. This article is about weakened (one-sided) involutions. As such, completely isotonic mappings are considered (they are defined in the second section). The properties of these mappings and their conditionally inverse mappings allow half-dual transitions - transitions in only one direction. Duality theorems claiming the possibility of such transitions are called one-sided duality schemes. The content of the work is an attempt to bring a unified mathematical base for all possible one-sided schemes of duality, which allows us to reformulate each of them in accordance with a single standard. This possibility is presented by the interpretation of dual transitions that arose under the conditions of the theory of spectral synthesis in the complex field as transitions from an injective (internal) description of some mathematical objects to a projective (external) description of other mathematical objects. The involutions used in one-sided schemes of duality, in turn, are one-sided and the restrictions imposed on them are much weaker. This leads to a significant expansion of the scope of the possible application of dual schemes in research practice.
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