Order based on associative operations

Inspired by the classical works on obtaining order from semigroups, recently, many researchers have proposed orders based on associative fuzzy logic connectives. However, the use of these monotone operators succinctly assumes and implies the presence of an (existing) order on the underlying set. In this work, we consider associative operations F on a non-empty set P without recourse to any ordering that may or may not be available on it. Picking the most general of the definitions of order proposed so far, that of Karaçal and Kesiciogˇlu, we determine the necessary and sufficient conditions on an associative operation F to obtain a poset on P. Following this we investigate the classes of t-norms, t-conorms, uninorms and nullnorms – which are the typical fuzzy logic operations considered so far to obtain orders – that satisfy these conditions and also do a comparative study of the structures obtained from the different orders proposed so far. Finally, we explore further conditions required on an associative operation F to obtain richer order-theoretic structures.