Betweenness relations and gated sets in fuzzy metric spaces

Abstract

Gated sets in metric spaces play an important role in the study of convexity. In this paper, we investigate the notion of gated sets in the context of betweenness of fuzzy metric spaces. For this purpose, we analyze the validity of postulates presented by Huntington and Kline to the betweenness in GV fuzzy metric spaces. The postulates are different from the usual properties of metric betweenness, amounting to thirteen. Interestingly, the validity of most of these postulates to the betweenness need to be considered in fuzzy normed spaces. In particular, two of the postulates for this relation are valid when the space is 1-dimensional. Five of the postulates for this relation are centrally studied in connection with strict convexity. As a consequence, several characterizations on strict convexity in a GV fuzzy normed space under the minimum t-norm are established.