A meadow is a commutative ring with an inverse operator satisfying 0⁻¹ = 0. We determine the initial algebra of the meadows of characteristic 0 and prove a normal form theorem for it. As an immediate consequence we obtain the decidability of the closed term problem for meadows and the computability of their initial object.

We apply a paraconsistent strategy to reasoning about fractions.

Inside a combinatory algebra, there are ‘internal’ versions of the finite type structure over ω, which form models of various systems of finite type arithmetic. This paper compares internal representations of the intensional and extensional functionals. If these classes coincide, the algebra is called ft-extensional. Some criteria for ft-extensionality are given and a number of well-known ca's are shown to be ft-extensional, regardless of the particular choice of representation for ω. In particular, DA, Pω, Tω, Hω and certain D∞-models all (...) share the property of ft-extensionality. It is also shown that ft-extensionality is by no means an intrinsic property of ca's i.e., that there exists a very concrete class of ca's—the class of reflexive coherence spaces—no member of which has this property. This leads to a comparison of ft-extensionality with the well-studied notions of extensionality and weak extensionality. Ft-extensionality turns out to be completely independent. (shrink)

The principal aim of this paper is to present a construction method for nontotal extensional combinatory algebras. This is done in $\S2$ . In $\S0$ we give definitions of some basic notions for partial combinatory algebras from which the corresponding notions for (total) combinatory algebras are obtained as specializations. In $\S1$ we discuss some properties of nontotal extensional combinatory algebras in general. $\S2$ describes a "partial" variant of reflexive complete partial orders yielding nontotal extensional combinatory algebras. Finally, $\S3$ deals with (...) properties of the models constructed in $\S2$ , such as incompletability, having no total submodel and the pathological behaviour with respect to the interpretation of unsolvable λ-terms. (shrink)

Adapting a claim of Kracht (Theor Comput Sci 354:131–141, 2006), we establish a characterization of the typable partial applicative structures.