Basic algebra of BK products of relations in t-norm fuzzy logics

Relational computing structures make it possible to perform knowledge representation as well as all the computations in intelligent systems in a unified way. When crisp computations are replaced by fuzzy relational computations, it is possible to improve significantly handling of indeterminacy and incompleteness of information. Fuzzy computational structures in which cuts commute with closures over relational properties, provide in addition the means for significant data compression, resulting in significant speedup of computations. BK-relational products provide axiomatics necessary for constructive computational procedures that rigorously satisfy the above formulated requirements. This paper provides the mathematical overview of concepts needed for fuzzy relational computations and the rigorous mathematical proofs of inequalities relating triangle BK-products to standard t-norm based compositions of relations. The proofs of correctness of these crucial computational relationships is performed in basic logic of Hajek (1996, 1998), which is the fuzzy predicate logic axiomatization of t-norm based residuated fuzzy logics. The paper concludes with a survey of the use of BK-products in scientific, medical, engineering and business applications, information retrieval, automated reasoning, etc.