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.     

Electricity market equilibrium based on conjectural variations

Many of the models used for the representation of the generation companies’ behavior in oligopolistic electricity markets are based on conjectural variation equilibria, which are a generalization of the classic Cournot approach. However, one of the main drawbacks of these approaches is the complexity of assigning adequate values to the conjectures, which can dramatically affect the results obtained from these models.     

A note on Trillas' CHC models

Trillas et al. [E. Trillas, S. Cubillo, E. Castiñeira, On conjectures in orthocomplemented lattices, Artificial Intelligence 117 (2000) 255-275] recently proposed a mathematical model for conjectures, hypotheses and consequences (abbr. CHCs), and with this model we can execute certain mathematical reasoning and reformulate some important theorems in classical logic. We demonstrate that the orthomodular condition is not necessary for holding Watanabe's structure theorem of hypotheses, and indeed, in some orthocomplemented but not orthomodular lattices, this theorem is still valid. We use the CHC operators to describe the theorem of deduction, the theorem of contradiction and the Lindenbaum theorem of classical logic, and clarify their existence in the CHC models; a number of examples is presented. And we re-define the CHC operators in residuated lattices, and particularly reveal the essential differences between the CHC operators in orthocomplemented lattices and residuated lattices.     

On the reducibility of hypotheses and consequences

This paper only tries to shed some additional light on the concepts of hypotheses and consequences, introduced by Trillas et al. [E. Trillas, S. Cubillo, and E. Castiñeira, On conjectures in orthocomplemented lattices, Artificial Intelligence, 117 (2) (2000) 255-275] as particular cases of the more general concept of conjecture. The definitions of reducible hypothesis and reducible consequence are presented, and it is shown that in orthomodular lattices all hypotheses and consequences are reducible, but that this is not necessarily the case in proper ortholattices. In addition, the sets of reducible hypotheses and consequences are characterized, both for general ortholattices as well as for the particular case of orthomodular lattices.