Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
quasigroup; composition of operations; orthogonal operations; perpendicular operations; hypercube; perpendicular hypercubes; orthogonality of hypercubes; slice; linear quasigroup; $T$-quasigroup
quasigroup; composition of operations; orthogonal operations; perpendicular operations; hypercube; perpendicular hypercubes; orthogonality of hypercubes; slice; linear quasigroup; $T$-quasigroup
Summary:
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.
References:
[1] Gluhov M.M.: About $\alpha$-closed classes and $\alpha$-complete systems of functions of $k$-valued logics. Discrete Math. 1 (1989), no. 1, 16–21. MR 1072636
[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). MR 1073918
[4] Belyavskaya G.: Pairwise orthogonality of $n$-ary operations. Bul. Acad. Stiinte Repub. Mold. Mat. 3 (2005), 5–18. MR 2225091 | Zbl 1148.20310
[5] Yurevych O.: About cross isotopy of poliagroup. Pratsi Instytutu prykladnoi matematyky i mechaniky NAN Ukraina 11 (2005), 34–39 (in Ukrainian). MR 2353778
[6] Belyavskaya G., Mullen G.L.: Orthogonal hypercubes and $n$-ary operations. Quasigroups Related Systems 13 (2005), no. 1, 73–86. MR 2206148 | Zbl 1101.20049
[7] Sokhatskyj F., Syvakivskyj P.: On linear isotopes of cyclic groups. Quasigroups Related Systems 1 (1994), no. 1, 66–76. MR 1327947
[8] Mullen G.L., Shcherbacov V.A.: On orthogonality of binary operations and squares. Bul. Acad. Stiinte Repub. Mold. Mat. 2 (2005), 3–42. MR 2190736
Search
Partner of
