#### Glasnik Matematicki, Vol. 37, No.2 (2002), 235-244.

### VARIETIES OF GRUPOIDS WITH AXIOMS OF THE FORM
*x*^{m+1}*y* =
*xy* AND/OR *xy*^{n+1} = *xy*

### Gorgi Čupona, Naum Celakoski and Biljana Janeva

Macedonian Academy of Sciences and Arts,
1000 Skopje, R. Macedonia

Faculty of Mechanical Engineering,
1000 Skopje, R. Macedonia

Faculty of Natural Sciences and Mathematics,
Institute of informatics,
PB 162, 1000 Skopje, R. Macedonia

*e-mail:* `biljana@pmf.ukim.edu.mk`

**Abstract.** The subject of this paper are varieties
U(*M*;*N*)
of groupoids defined by the following system of identities

{ *x*^{m+1}*y* = *xy* : *m*
∈ *M* }
∪
{ *x**y*^{n+1} = *xy* :
*n* ∈ *N* },

where *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 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.

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