Quasigroups, loops, and associative laws. (English) Zbl 0860.20053

The author investigates the question of which weakenings of the associative law imply that a quasigroup is a loop. In particular, he completely settles the question for all laws which are written with four variables, three of which are distinct (“size four laws”). In earlier work [J. Algebra 183, No. 1, 231-234 (1996; Zbl 0855.20056)], he had shown that every quasigroup satisfying any one of the four Moufang laws is a loop. In the paper under review he provides now a wealth of information on size four laws, culminating in theorems stating that, other than laws which are equivalent to the full associative law, each of the following laws and their “mirrors” (obtained by writing the equation backwards), and each of the Moufang laws, are those whose validity implies that a quasigroup is a loop: \[ \text{(i) }(x(xy))z=(xx)(yz),\quad\text{(ii) }(x(yx))z=(xy)(xz),\quad\text{(iii) }(x(yy))z=x((yy)z), \]
\[ \text{(iv) }(x(yy))z=(xy)(yz),\quad\text{(v) }(x(yz))y=(xy)(zy),\quad\text{(vi) }((xy)z)x=x(y(zx)). \] According to the author, his investigations have been aided by the automated deduction tools OTTER, FINDER, and MACE.
Reviewer: R.Artzy (Haifa)


20N05 Loops, quasigroups
20A05 Axiomatics and elementary properties of groups


Zbl 0855.20056
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