INFORMATION AND CONTROL 24, 55-73 (1974) Entropy of L-Fuzzy Sets* ALDO DE LUCA AND SETTIMO TERMINI Laboratorio di Cibernetica del Consiglio Nazionale delle Ricerche, Arco Felice, Napoli, Italy The notion of "entropy" of a fuzzy set, introduced in a previous paper in the case of generalized characteristic functions whose range is the interval [0, 1] of the real line, is extended to the case of maps whose range is a poset L (or, in particular, a lattice). Some of the reasons giving rise to the non-comparability of the truth values and then the necessity of considering poset structures as range of the maps are discussed. The interpretative problems of the given mathematical definitions regarding the connections with decision theory are briefly analyzed. 1. INTRODUCTION In this work the notion of entropy of a fuzzy set (Zadeh, 1965) introduced in De Luca and Termini (1972) will be extended to the case of L-fuzzy sets (Goguen, 1967), i.e., maps from a given set I to a partly ordered set (poset) L. The difference between the case in which L coincides with the interval [0, 1] of the real line (fuzzy sets) and that considered here, in which L is a general poset, is similar to the one existing between a multivalued logic (see, for instance, Lukasiewicz and Tarski, 1930) and a logic in which the truth- values of the propositions are not always comparable (see, for instance, Koopman, 1940). In the next section we shall collect some mathematical preliminaries that will be used in the following. Before introducing the formal definition of entropy of an L-fuzzy set, we shall, then, briefly discuss how and why the necessity arises of considering poset structures as range of the generalized characteristic functions. The mathematical definition of the entropy of an L-fuzzy set will then be given in the general case. This entropy is a matrix quantity so that the L-fuzzy sets cannot always be compared by means of their entropies. A more detailed * An abstract of this paper was published in Notices A.M.S. 19, A710 (1972). 55 Copyright © 1974 by Academic Press, Inc. All rights of reproduction in any form reserved.