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Algorithm for the complement of orthogonal operations

Iryna V. Fryz — 2018

Commentationes Mathematicae Universitatis Carolinae

G. B. Belyavskaya and G. L. Mullen showed the existence of a complement for a $k$ -tuple of orthogonal $n$ -ary operations, where $k < n$ , to an $n$ -tuple of orthogonal $n$ -ary operations. But they proposed no method for complementing. In this article, we give an algorithm for complementing a $k$ -tuple of orthogonal $n$ -ary operations to an $n$ -tuple of orthogonal $n$ -ary operations and an algorithm for complementing a $k$ -tuple of orthogonal $k$ -ary operations to an $n$ -tuple of orthogonal $n$ -ary operations. Also we find some...

Invertibility criterion of composition of two multiary quasigroups

Fedir M. Sokhatsky; Iryna V. Fryz — 2012

Commentationes Mathematicae Universitatis Carolinae

We study invertibility of operations that are composition of two operations of arbitrary arities. We find the criterion for quasigroups and specifications for $T$ -quasigroups. For this purpose we introduce notions of perpendicularity of operations and hypercubes. They differ from the previously introduced notions of orthogonality of operations and hypercubes [Belyavskaya G., Mullen G.L.: Orthogonal hypercubes and $n$ -ary operations, Quasigroups Related Systems 13 (2005), no. 1, 73–86]. We establish...

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