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) is an element of A(3), 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 F G -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. (c) 2004 Elsevier B.V. All rights reserved.