Varieties of grupoids with axioms of the form x^{m+1}y = xy and/or xy^{n+1} = xy

The subject of this paper are varieties (M;N) of groupoids defined by the following system of identities { xm+1y = xy : m M } { xyn+1 = xy : n N }, where M, N are sets of positive integers. The equation (M;N) = (M\u27;N\u27) for any given pair (M,N) is solved, and, among all solutions, one called canonical, is singled out. Applying a result of Evans it is shown for finite M and N that: if M and N are nonempty and gcd(M) = gcd(M N), or only one of M and N is nonempty, then the word problem is solvable in (M;N)

Varieties of grupoids with axioms of the form x^{m+1}y = xy and/or xy^{n+1} = xy

The subject of this paper are varieties (M;N) of groupoids defined by the following system of identities { xm+1y = xy : m M } { xyn+1 = xy : n N }, where M, N are sets of positive integers. The equation (M;N) = (M\u27;N\u27) for any given pair (M,N) is solved, and, among all solutions, one called canonical, is singled out. Applying a result of Evans it is shown for finite M and N that: if M and N are nonempty and gcd(M) = gcd(M N), or only one of M and N is nonempty, then the word problem is solvable in (M;N)

Around the Hossz\'u-Gluskin theorem for nn-ary groups

We survey results related to the important Hossz\'u-Gluskin Theorem on $n$-ary groups adding also several new results and comments. The aim of this paper is to write all such results in uniform and compressive forms. Therefore some proofs of new results are only sketched or omitted if their completing seems to be not too difficult for readers. In particular, we show as the Hossz\'u-Gluskin Theorem can be used for evaluation how many different $n$-ary groups (up to isomorphism) exist on some small sets. Moreover, we sketch as the mentioned theorem can be also used for investigation of $\mathcal{Q}$-independent subsets of semiabelian $n$-ary groups for some special families $\mathcal{Q}$ of mappings