Relations among some classes of quasigroups

We show relations among some classes of quasigroups. A quasigroup (G,.) is calIed unipotent if it contains an element x  such that a.a = x, for every a in G; subtractive  if b.(b.a) = a and a. (b.c) = c|.(b.a)  for all a,b,c in G; medial if (a.b).(c.d) =(a.c).(b.d) for all a,b,c,d in G. We define a Ward quasigroup as any quasigroup (G,.) containing an element i ϵ G such that a.a = i and (a.b).c = a.(c.(i.b)) for all a,b,c in G. A quasigroup (G,.) which contains an element i that satisfies the axioms a.x = b ↔ x = (i.b).(i.a) and y.a = b↔ y = b.(i.a) for all a,b in G, is called a Cardoso quasigroup by A. Sade [4]. If the class of the quasigroup is denoted by the initial letter of the respective name, then:(1) S ⊂W⊂C⊂U; (2) M∩C = S. There is no relation of inclusion between the class of loops and any of the other classes; we exhibit examples to evidence this fact. Furthermore, we establish necessary and sufficient conditions for a Cardoso quasigroup to be a loop.

This paper is concerned with relations among the classes of the following quasigroups: subtractive, medial, Cardoso, Ward, unipotent and loop. In a sense, it is a continuation of [5], where some types of unipotent quasigroups were studied. Ward quasigroups are important because there is a conection between these quasigroups and groups [2]. Namely, if (G,.) is a Ward quasigroup, then (G,*) is a group under the operation * defined by a*b = a.(i.b); conversely, if (G,*) is agroup, then (G,_.) is a Ward quasigroup with respect to the operation  defined by a.b = a*b^-1. In particular, if the

group (G,*) is abelian, then the quasigroup is subtractive.