Crossed-inverse and related loops

Introduction. Crossed-inverse (C.I.) loops are loops in which the identity xy x'=y holds for all x, y of the loop, x' being the right inverse of x. Such loops have already been studied [1; 2], and the results were useful in the study of neofields [6] and of geometrical subjects [3]. C.I. loops are special automorphic-inverse loops, that is, loops satisfying the identity (xy)'=x'y', for all x, y in the loop. The first section of this paper deals with isotopic C.I. loops, and they are proved to be also isomorphic. Then the autotopisms of C.I. loops are studied, and the main result of this part is the fact that the "companions" of all the autotopisms, i.e. the elements that correspond to the translation elements of the isotopisms, form a subloop which is commutative and Moufang. ?2 of the paper gives a necessary condition for the existence of finite automorphicinverse loops consisting only of the unit eleinent and inverse-cycles [cf. 1 ] of the same length. The condition concerning the equal length of all inversecycles is necessary for the transitivity of the automorphism group of the loop [2] and hence for the possibility of considering the loop as the additive loop of a neoring. The third part deals with a method of constructing infinite automorphic-inverse loops satisfying also certain other identities. The author is grateful to R. H. Bruck for fruitful discussions on the subject of this paper. 1. 1. Definitions. 1.11. The biunique mapping which maps every element, c, of a loop (G, *) on its right inverse will be called J. Thus c cJ= 1, and cJ-1 c= 1. In automorphic-inverse loops J is an automorphism. 1.12. In the principal isotope of the loop G, with the new operation (*) such that ag*fb =ab, for all a, b in G, the ordered pair (g, f) is called the pair of translation elements of the principal isotope. 1.2. Elementary properties of C.I. loops. 1.21. The identities ab aJ=b and a(b aJ) =b are equivalent. Proof. If P and Q are biunique mappings of the loop onto itself, then PQ= I, the identity mapping, if and only if QP=I. Let L(x) and R(x) be,