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ЖУРНАЛЫ // Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica // Архив

Bul. Acad. Ştiinţe Repub. Mold. Mat., 2023, номер 3, страницы 103–106 (Mi basm606)

Short communications

On the order of recursive differentiability of finite binary quasigroups

Parascovia Syrbu

Moldova State University, Department of Mathematics

Аннотация: The recursive derivatives of an algebraic operation are defined in [1], where they appear as control mappings of complete recursive codes. It is proved in [1], in particular, that the recursive derivatives of order up to $r$ of a finite binary quasigroup $(Q,\cdot )$ are quasigroup operations if and only if $(Q,\cdot )$ defines a recursive MDS-code of length $r+3$. The author of the present note gives an algebraic proof of an equivalent statement: a finite binary quasigroup $(Q,\cdot )$ is recursively $r$-differentiable $(r\geq 0)$ if and only if the system consisting of its recursive derivatives of order up to $r$ and of the binary selectors, is orthogonal. This involves the fact that the maximum order of recursive differentiability of a finite binary quasigroup of order $q$ does not exceed $q-2$.

Ключевые слова и фразы: quasigroup, recursive derivative, recursively differentiable quasigroup.

MSC: 20N05, 20N15, 11T71

Поступила в редакцию: 21.07.2021

Язык публикации: английский

DOI: 10.56415/basm.y2023.i3.p103



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