Finite irredundant systems of identities in commutative Moufang loops and distributive Steiner quasigroups

N. I. Sandu

Izvestiya. Mathematics , Tome 30 (1988) no. 1, pp. 163-183

Résumé

An infinite irredundant system of identities in the variety of commutative Moufang loops (and in the variety of distributive Steiner quasigroups) is constructed with the property that no infinite subset is equivalent to a finite subsystem. It follows that the set of subvarieties of the variety of commutative Moufang loops (and of the variety of distributive Steiner quasigroups) has the cardinal of the continuum; it also follows that there is a commutative Moufang loop (a distributive Steiner quasigroup) given by an enumerable set of identical relations, in which the word problem is unsolvable. Finally, the three arguments conjecture is refuted for commutative Moufang loops. In fact, an associator is constructed that is not the identity but has three of its arguments the same. Bibliography: 10 titles.