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  (inspired by E. N¨other). The coset theorem of Post explained the connection between n -ary groups and their covering binary groups POST . The next step in study of n -ary groups was the Gluskin-Hossz ´u theorem HOSSZ´ U, GLUSKIN. The cubic and n -ary generalizations of matrices and determinants were made in KAPRANOV ET AL. , SOKOLOV , physical application in KAWAMURA , RAUSCH DE TRAUBENBERG.