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

Monotonicity of SISO Fuzzy Relational Inference With an Implicative Rule Base

A fuzzy relational inference (FRI) mechanism is appraised based on the different desirable properties it possesses. Among these properties, monotonicity of an FRI has not received much attention. In this work, we investigate the monotonicity of a single-input single-output FRI with an implicative form of the rule base. In all the previous works that deal with monotonicity of an FRI with the implicative form of rule base, the employed fuzzy implications come from a residuated lattice. It can be noticed that this rich underlying structure plays a major role in proving the results. Further, we also modify the given monotone rule base. This work differs from the previous works in that: 1) the fuzzy implications employed in it do not come from any known residuated structure on [0, 1]; and 2) the original rule base is employed without any alteration. We determine conditions under which monotonicity of an FRI, where the rule base is modeled by a strict fuzzy implication, can be ensured without transforming the original rule base. Thus, the results in this work further augment the case for considering fuzzy implications, other than those from the residuated setting, to be used in applications