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)