1 History Ternary algebraic operations (with the arity n = 3 ) were introduced by A. Cayley in 1845 and later by J. J. Sylvester in 1883. The notion of an n -ary group was introduced in 1928 by D ¨ ORNTE [1929] (inspired by E. N¨other). The coset theorem of Post explained the connection between n -ary groups and their covering binary groups P OST [1940]. The next step in study of n -ary groups was the Gluskin-Hossz ´u theorem H OSSZ ´ U [1963], G LUSKIN [1965]. The cubic and n -ary generalizations of matrices and determinants were made in K APRANOV ET AL . [1994], S OKOLOV [1972], physical application in K AWAMURA [2003], R AUSCH DE T RAUBENBERG [2008].

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