Transitivity frameworks for reciprocal relations: cycle-transitivity versus FG-transitivity

For a reciprocal relation Q on a set of alternatives A, two transitivity frameworks which generalize both T-transitivity and stochastic transitivity are compared: the framework of cycle-transitivity, introduced by the present authors (Soc. Choice Welf., to appear) and which is based upon the ordering of the numbers Q(a,b), Q(b,c) and Q(c,a) for all (a,b,c)∈A3, and the framework of FG-transitivity, introduced by Switalski (Fuzzy Sets and Systems 137 (2003) 85) as an immediate generalization of stochastic transitivity. The rules that enable to express FG-transitivity in the form of cycle-transitivity and cycle-transitivity in the form of FG-transitivity, illustrate that for reciprocal relations the concept of cycle-transitivity provides a framework that can cover more types of transitivity than does the concept of FG-transitivity.