ALL COMPACT HAUSDORFF LAMBDA MODELS ARE DEGENERATE

Article type: Research Article

Authors: Hofmann, K.H. | Mislove, M.W.

Affiliations: Fachbereich Mathematik, Schloßgartenstraße 7, D-64289 Darmstadt, Germany | Department of Mathematics, Tulane University, New Orleans, LA 70118

Note: [] The first author thanks Deutsche Forschungsgemeinschaft and the second the Office of Naval Research for their support. E-mail addresses: first author: hofmann@mathematik.th-darmstadt.de, and second author: mwm@tulmath.math.tulane.edu

Note: [] The first author thanks Deutsche Forschungsgemeinschaft and the second the Office of Naval Research for their support. E-mail addresses: first author: hofmann@mathematik.th-darmstadt.de, and second author: mwm@tulmath.math.tulane.edu

Abstract: The first mathematical model of untyped lambda calculus was discovered by DANA SCOTT in the category of algebraic lattices and Scott continuous maps. The question then arises as to which other cartesian closed categories contain a model of the calculus. In this paper we show that any compact Hausdorff model of the calculus must satisfy the property that the semantic map from the calculus to the model is constant. In particular, any compact reflexive object in the category of Hausdorff k-spaces gives rise to a degenerate model of the calculus. We also explore the relationship of the results we derive to the notions of a combinatory model and of an environment model of the calculus.

DOI: 10.3233/FI-1995-22122

Journal: Fundamenta Informaticae, vol. 22, no. 1-2, pp. 23-52, 1995