Showing papers in "Journal of The London Mathematical Society-second Series in 1976"

A Syntactic Characterization of the Equality in Some Models for the Lambda Calculus

Abstract: An equality relation on the terms of the A-calculus is an equivalence relation closed under the (syntactical) operations of application and A-abstraction. We may distinguish between syntactic and semantic ways of introducing equality relations, /^-equality is introduced syntactically; it is the least equality relation satisfying the equations for aand ^-conversion. For a more subtle way of introducing equality relations syntactically, consider the relations =f and =h of §5 of this paper. To give a semantic characterization of an equality relation, we simply take the relation ' has the same value in £>', where D is some model for the A-calculus. Of course, no equality relation is of interest to the intended interpretation of the A-calculus, unless it extends /^-equality. An equality relation is inconsistent if and only if it sets all terms equal; otherwise it is consistent. It is maximal consistent if and only if it is consistent and has no consistent proper extensions. In this paper we consider a class of continuous lattice models for the A-calculus, and a particular model, the Graph model. The same equality is induced by all the continuous lattice models; we shall refer to them as the Scott models (see [3], where they were first constructed). For the history of the Graph model see [4]. We shall give, in this paper, syntactic characterizations of the equality induced by the Scott models, and by the Graph model; and we shall show that the equality induced by the Scott models is the unique maximal consistent equality relation, extending the relation = H, which was proved consistent in [1]. We use x, y,z, w ... for variables, and M,N, P ... for terms of the A-calculus (with subscripts as necessary). D will refer to whatever model or models are under consideration. The content of our Theorem 5.4 (a) has been discovered independently by C. P. Wadsworth.

On the structure of edge graphs ii

Abstract: This note is a sequel to [1]. First let us recall some of the notations. Denote by G{n, m) a graph with n vertices and m edges. Let Kd(ru ..., rd) be the complete dpartite graph with r{ vertices in its i-th class and put Kd(t) = Kd(t, ..., t), Kd = Kd(\\). Given integers n ̂ d(^ 2), let md(n) be the minimal integer with the property that every G(n, m), where m ^ md(ri), contains a Kd. The function md(n) was determined by Turan [5]. It is easily seen that

Special Linear Systems on Curves Lying on a K 3 Surface

Abstract: In this article C will aways denote a nonsingular curve of genus g lying on a K3 surface X. By a g1 r I understand a linear system of degree r and dimension 1 which is without fixed points and complete. The g1 r is said to be separable if the associated map to P1 is, and this is obviously equivalent to the g1 r containing a divisor P1 + · · ·+ Pr made up of distinct points Pi. My aim is to prove the following result.

Disjoint Common Transversals and Exchange Structures

Abstract: We state and prove a theorem (Theorem 1 below) which strengthens previously known results concerning disjoint common partial transversals of two families of sets. This theorem may be viewed as a result on transversal pre-independence structures. We define a \"disjoint-exchange structure\" on a set and extend the result to such structures (Theorem 3 below). Then we give an application of this theorem to strongly-base-orderable matroids, and deduce a result on the existence of a number of common transversals of two families of sets with each pair of transversals intersecting in a given set.