Abstract. In this paper we describe a language and method for deriving ontologies and ordering databases. The ontological structures arrived at are distributive lattices with attribution operations that preserve ∨, ∧ and ⊥. The preservation of ∧ allows the attributes to model the natural join operation in databases. We start by introducing ontological frameworks and knowledge bases and define the notion of a solution of a knowledge base. The import of this definition is that it specifies under what condition all information relevant to the domain of interest is present and it allows us to prove that a knowledge base always has a smallest, or terminal, solution. Though universal or initial solutions almost always are infinite in this setting with attributes, the terminal solution is finite in many cases. We describe a method for computing terminal solutions and give some conditions for termination and non-termination. The approach is predominantly coalgebraic, using Priestley duality, and calculations are made

Abstract. We use coalgebraic methods to describe finitely generated free Heyting algebras. Heyting algebras are axiomatized by rank 0-1 axioms. In the process of constructing free Heyting algebras we first apply existing methods to weak Heyting algebras—the rank 1 reducts of Heyting algebras—and then adjust them to the mixed rank 0-1 axioms. On the negative side, our work shows that one cannot use arbitrary axiomatizations in this approach. Also, the adjustments made for the mixed rank axioms are not just purely equational, but rely on properties of implication as a residual. On the other hand, the duality and coalgebra perspective does allow us, in the case of Heyting algebras, to derive Ghilardi’s (Ghilardi, 1992) powerful representation of finitely generated free Heyting algebras in a simple, transparent, and modular way using Birkhoff duality for finite distributive lattices. 1

a logic step-by-step using duality

In this paper we introduce a new setting, based on partial algebras, for studying constructions of finitely generated free algebras. We give sufficient conditions under which the finitely generated free algebras for a variety V may be described as the colimit of a chain of finite partial algebras obtained by repeated application of a functor. In particular, our method encompasses the construction of finitely generated free algebras for varieties of algebras for a functor as in [2], Heyting algebras as in [1] and S4 algebras as in [8]. 1