Abstracting from certain properties of the implication operation in Boolean algebras leads to so-called orthoimplication algebras. These are in a natural one-to-one correspondence with families of compatible orthomodular lattices. It is proved that congruence kernels of orthoimplication algebras are in a natural one-to-one correspondence with families of compatible p-filters on the correspondin...

The implication of multivalued dependencies (MVDs) in relational databases has originally been defined in the context of some fixed finite universe. While axiomatisability and implication problem have been intensely studied with respect to this notion, almost no research has been devoted towards the alternative notion of implication in which the underlying universe of attributes is left undeter...

C. I. Lewis repeatedly exempts MacColl from criticisms of his predecessors in their accounts of implication. They had all taken a true implication, or conditional, to be one with false antecedent or true consequent. MacColl uniquely, and correctly in Lewis’ view, rejected this account, identifying a true implication with the impossibility of true antecedent and false consequent. Lewis’ developm...

In this paper, we study the relationships between left (right) semi-uninorms and implications on a complete lattice. We firstly discuss the residual operations of left and right semi-uninorms and show that the right (left) residual operator of a conjunctive right (left) ∨-distributive left (right) semi-uninorm is a right ∧-distributive implication that satisfies the neutrality principle. Then, ...

Functional dependencies (FDs) and inclusion dependencies (INDs) are the most fundamental integrity constraints that arise in practice in relational databases. We introduce null inclusion dependencies (NINDs) to cater for the situation when a database is incomplete and contains null values. We show that the implication problem for NINDs is the same as that for INDs. We then present a sound and c...