Invertibility criterion of composition of two multiary quasigroups

Fedir M. Sokhatsky; Iryna V. Fryz

Commentationes Mathematicae Universitatis Carolinae (2012)

  • Volume: 53, Issue: 3, page 429-445
  • ISSN: 0010-2628

Abstract

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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 some relationships between these notions and give illustrative examples.

How to cite

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Sokhatsky, Fedir M., and Fryz, Iryna V.. "Invertibility criterion of composition of two multiary quasigroups." Commentationes Mathematicae Universitatis Carolinae 53.3 (2012): 429-445. <http://eudml.org/doc/246163>.

@article{Sokhatsky2012,
abstract = {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 some relationships between these notions and give illustrative examples.},
author = {Sokhatsky, Fedir M., Fryz, Iryna V.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {quasigroup; composition of operations; orthogonal operations; perpendicular operations; hypercube; perpendicular hypercubes; orthogonality of hypercubes; slice; linear quasigroup; $T$-quasigroup; compositions of operations; orthogonal operations; perpendicular operations; perpendicular hypercubes; linear quasigroups; -quasigroups},
language = {eng},
number = {3},
pages = {429-445},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Invertibility criterion of composition of two multiary quasigroups},
url = {http://eudml.org/doc/246163},
volume = {53},
year = {2012},
}

TY - JOUR
AU - Sokhatsky, Fedir M.
AU - Fryz, Iryna V.
TI - Invertibility criterion of composition of two multiary quasigroups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2012
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 53
IS - 3
SP - 429
EP - 445
AB - 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 some relationships between these notions and give illustrative examples.
LA - eng
KW - quasigroup; composition of operations; orthogonal operations; perpendicular operations; hypercube; perpendicular hypercubes; orthogonality of hypercubes; slice; linear quasigroup; $T$-quasigroup; compositions of operations; orthogonal operations; perpendicular operations; perpendicular hypercubes; linear quasigroups; -quasigroups
UR - http://eudml.org/doc/246163
ER -

References

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  1. Gluhov M.M., About α -closed classes and α -complete systems of functions of k -valued logics, Discrete Math. 1 (1989), no. 1, 16–21. MR1072636
  2. Sokhatsky F.M., The deepest repetition-free decompositions of non-singular functions of finite valued logics, Proceedings of The Twenty-Sixth International symposium on Multiple-Valued Logic (Santiago de Compostela, Spain, May 29-31, 1996), pp. 279–282. 
  3. Belousov V.D., Crossed isotopies of quasigroup, Quasigroups and their Systems, Stiintsa, Chishinau, 1990, 14–20 (in Russian). MR1073918
  4. Belyavskaya G., Pairwise orthogonality of n -ary operations, Bul. Acad. Stiinte Repub. Mold. Mat. 3 (2005), 5–18. Zbl1148.20310MR2225091
  5. Yurevych O., About cross isotopy of poliagroup, Pratsi Instytutu prykladnoi matematyky i mechaniky NAN Ukraina 11 (2005), 34–39 (in Ukrainian). MR2353778
  6. Belyavskaya G., Mullen G.L., Orthogonal hypercubes and n -ary operations, Quasigroups Related Systems 13 (2005), no. 1, 73–86. Zbl1101.20049MR2206148
  7. Sokhatskyj F., Syvakivskyj P., On linear isotopes of cyclic groups, Quasigroups Related Systems 1 (1994), no. 1, 66–76. MR1327947
  8. Mullen G.L., Shcherbacov V.A., On orthogonality of binary operations and squares, Bul. Acad. Stiinte Repub. Mold. Mat. 2 (2005), 3–42. MR2190736

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