Linearity and linear extensions of fuzzy orders revisited

Abstract

The famous Szpilrajn's extension theorem demonstrating existence of linear extension of any order is one of the most important results in order theory. Inevitably its various generalizations were studied also in the setting of fuzzy logic, where up to date results are rather pessimistic. Focusing on the arguably most developed approach to fuzzy orders where order relation is defined with respect to a fuzzy equality already present within the universe, we reevaluate the strength of a link between such fuzzy order and the underlying equality. We first observe that compared to the Boolean setting, the situation is significantly more interesting in the setting of fuzzy logic as there may be many fuzzy equalities on the given set. Then we show that the link is in a sense more substantial than usually assumed and should be considered in both directions. That is defining fuzzy order with respect to fuzzy equality is not enough, the fuzzy equality should moreover mirror all the adjustments made to the fuzzy order accordingly. Utilizing this observation, we provide a generalization of Szpilrajn's extension theorem within the framework of fuzzy logic, which further alleviates the drawbacks that comparable generalizations possessed in previous studies.