Ivashchuk O. Multiary associativity and related groupoids

Three kinds of axiom systems are found: in terms of invertible elements; in terms of neutral operations; and in terms of neutral, symmetric and mirrow operations. The well known axiomatics of n-ary groups are obtained as corollaries. A serie of element invertiblity criteria in associates are proved. Properties of cross isotopy and cross isomorphism of polyagroups are established.The structure of cross isotopies (autotopies) of medial polyagroups and the structure of segmental cross isotopies (autotopies) of polyagroups are found as well as the structure of strong cross isomorphisms of medial polyagroups. Some coinsiding conditions for isomorphisms and cross isomorphisms are stated. Some conditions for a cross isomorph of a polyagroup to be commutative or medial are found as well as a cross isotope of a quasigroup to be a quasigroup.