Geodesic
On extensions of partial $n$-quasigroups of order~4
Matematičeskie trudy, Tome 14 (2011) no. 2, pp. 147-172

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We prove that every collection of pairwise compatible (nowhere coinciding) $n$-ary quasigroups of order 4 can be extended to an $(n+1)$-ary quasigroup. In other words, every Latin $4\times\cdots\times4\times l$-parallelepiped, where $l=1,2,3$, can be extended to a Latin hypercube. 
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     author = {V. N. Potapov},
     title = {On extensions of partial $n$-quasigroups of order~4},
     journal = {Matemati\v{c}eskie trudy},
     pages = {147--172},
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     year = {2011},
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     url = {http://geodesic.mathdoc.fr/item/MT_2011_14_2_a6/}
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V. N. Potapov. On extensions of partial $n$-quasigroups of order~4. Matematičeskie trudy, Tome 14 (2011) no. 2, pp. 147-172. http://geodesic.mathdoc.fr/item/MT_2011_14_2_a6/