Consequences and conjectures in preordered sets

In a preordered set, or preset, consequence operators in the sense of Tarski, defined on families of subsets, are introduced. From them, the corresponding sets of conjectures, hypotheses, speculations and refutations are considered, studying the relationships between these sets and those previously defined on ortholattices. All the concepts introduced are illustrated with three particular consequence operators, whose behavior is studied in detail. The results obtained are applied to the case of fuzzy sets endowed with the usual pointwise ordering.