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By Ol'shanskii A. Y.
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Additional resources for A. I. Maltsevs problem on operations on groups
Qn ) are elements of Mlt Q. Then for each q in Q, one has EQ (p1 , . . , pm ) = FQ (q1 , . . , qn ) ⇒ qEQ (p1 , . . , pm ) = qFQ (q1 , . . , qn ) ⇒ wE (q, p1 , . . , pm ) = wF (q, q1 , . . , qn ) ⇒ wE (q V , pV1 , . . , pVm ) = wF (q V , q1V , . . , qnV ) ⇒ EQV (pV1 , . . , pVm ) = FQV (q1V , . . , qnV ). 11) slightly, one obtains a combinatorial multiplication group functor Mlt from the category of surjective quasigroup homomorphisms to the category of group epimorphisms, taking a morphism f : P → Q to Mlt f : Mlt P → Mlt Q; EP (p1 , .
46) w1 with diverging paths. It will be shown that one of the following occurs: Triangle: There is a chain of reductions from one of w1 , w1 to the other, without loss of generality from w1 to w1 : w1 → · · · → w1 . In this case w = w1 . Diamond: There is a word w0 in W that lies on reduction chains w1 → · · · → w0 from w1 and w1 → · · · → w0 from w1 . In this case w = w0 . Suppose that w = uvµg for words u, v in W . A reduction w → w1 is said to be internal if it is of the form uvµg → u1 vµg for a reduction u → u1 of u, or else of the form uvµg → uv1 µg for a reduction v → v1 of v.
Such an extension Q is said to be free if the embedding of X in any extension Q extends to a unique quasigroup homomorphism from Q to Q . The goal of this section is to show that each partial Latin square (X, U ) possesses a free extension Q(X,U ) , and to give an explicit description of the extension. Let (X, U ) be a partial Latin square. 40) of binary operations, satisfying the hypercommutative and hypercancellation laws. 40). 40) — or more precisely its image in the disjoint union — acts as a set of binary operations on (X + µS3 )∗ , with µg : (w, w ) → ww µg for w, w in (X + µS3 )∗ and g in S3 .