Varieties of grupoids with axioms of the form x^{m+1}y = xy and/or xy^{n+1} = xy
Gorgi Cupona, Naum Celakoski, Biljana Janeva · University of Zagreb University Computing Centre (SRCE) · 2002
The subject of this paper are varieties U (M ; N ) of groupoids defined by the following system of identitieswhere M, N are sets of positive integers.The equation U (M ; N ) = U (M ; N ) for any given pair (M, N ) is solved, and, among all solutions, one called canonical, is singled out.Applying a result of Evans ([6]) 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 U (M ; N ).2000 Mathematics Subject Classification.03C05, 03D40, 08A50, 08A55, 08B20.Key words and phrases.Groupoids, varieties of groupoids, partial groupoids, free groupoids, word problem.1 gcd(M ) is the greatest common divisor of M 2 N is the additive groupoid of integers generated by N .3 V |= τ 1 = τ 2 means: the equation τ 1 = τ 2 is true in the variety V. 4 m|n denotes that m is a divisor of n. 5 min(M ) denotes the least element in M .