Order from non-associative operations

Algebraic structures are often converted to ordered structures to gain information about the algebra using the properties of partially ordered sets. Such studies have been predominantly undertaken for semigroups, using various proposed relations. This has led to a spate of works dealing with associative fuzzy logic connectives (FLCs) and the orders that they generate. One such relation, proposed by Clifford, is employed both for its generality as well as utility. In a recent work, Gupta and Jayaram classified the semigroups that yield a partial order through the relation. In this work, we characterise groupoids that would give a partial order by introducing a property called the Generalised Quasi-Projectivity. Further, for the groupoids that lead to an ordered set, we explore the monotonicity of the underlying groupoid operation on the obtained poset. Finally, in light of the above results, we explore the major non-associative fuzzy logic connectives along these lines, thus complementing and augmenting, already existing works in the literature. Our work also shows when an FLC from a given class of operations remains one even w.r.to the order generated from it.