Note on Invariant Complexes of a Quasigroup

Abstract

In an earlier paper [1, ?4]2 the term invariant complex wN-as applied to any subset II of the elements of a finite quasigroup G such that (Ha) (Hb) for all a, b takes the form Hc for some element c in G. It was shown that a necessary and sufficient condition for the existence of a given homomorphism of G is that G contain an invariant complex H. MAore specifically, a homomorphism partitions the elements of G into disjoint subsets (complexes) H1 , H2 , such that any one of them may be denoted by H and then [1, Theorem 4.14] a-' of them, H included, take the form Hp for suitably chosen elements p of G. These complexes, under the rule of operation (Ha) (Hb) = He constitute a quasigroup (the quotient quasigroup G/H) isomorphic with the homomorph of G. Conversely, an invariant complex has the property that for its cosets Hx we have Ha = Hb or Ha A Hb = 0 so that, if p < Ha, the mapping p -* Ha constitutes a homomorphism of G. In view of the preceeding remarks, it is clear that a study of the relations H < K (proper inclusion of H by K), H A K = D, or HK = M between invariant complexes H and K is fundamental to the theory of homomorphisms of G. In [1, ??4.4, 4.5] such a study was undertaken, but with restrictions on H and K of the type (Hx) (Hy) = H(xy) or, still stronger, H(Ha) = Ha for all a. In Section 2 of the present paper the above relations are studied without the restrictions mentioned. If D t 0, D and M are shown (Theorems 2.1 and 2.4) to be invariant under no stronger hypothesis than that H and K are invariant. Theorem 2.2 shows the implications of H < K and Theorem 2.3 those of Hp A Kq # 0 for cosets of H and K. In Section 3 properties of maximal invariant complexes are studied in the light of Section 2. One of the principal difficulties encountered is the fact that none, one, or more than one of the cosets of an invariant complex may be a subquasigroup. This means that the analogy between invariant complexes of quasigroups and invariant subgroups of groups is not as close in Section 3 as in Section 2. However, if it be required, as it is in Section 3.2, that the quasigroup contain at least one idempotent element b considerable similarity of results remains. For example, Theorem 3.3 on composition series which terminate with the idempotent element b is an analogue of the Jordan-Holder Theorem. For a loop (a quasigroup with a two sided identity element e) introduced by A. A. Albert [2] there is only one idempotent element, namely e. The theory