On Cartesian monoids

  • Rick Statman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1258)


A Cartesian monoid is a structure (M, *, I, L, R, 〈<>〉) where (M, *, I) is a monoid with L and R members of M and 〈〉: M2M s.t. L * 〈x, y〉=x R * 〈x, y〉=xx, y〉 * z=〈x * z, y * z〉 〈L, R〉=I.

Cartesian monoids are easy to come by. Any surjective pairing function lifts pointwise to a 〈〉 on the monoid M of all functions under composition with L and R the lifted projections. Cartesian monoids were first introduced by Dana Scott in [5] and independently by Lambek in [3]. We first learned about them from Dana and Peter Freyd. There are many connections to the lambda calculus. First, the connection to simply typed lambda calculus with surjective pairing is transparent and forms the basis for [7]. Second, such monoids always contain a copy of the Freyd-Heller group (see below) thus there is a further connection to lambda calculus ([6], [10]). Finally, such monoids come up in the study of type algebras especially in connection to Curry's subject reduction theorem ([8], [9]).

It is the purpose of this paper to collect in one place pur observations on Cartesian monoids, especially the free Cartesian monoid, which are useful for applications to lambda calculus. In particular, we shall derive the undecidability of the matching and unification problems for F (in a very strong form). Our approach is to treat the free Cartesian monoid as an algebraic structure of the sort that we learned about in school. This is not to say that we have any argument with the category theoretic approach; it is only to say that we are not competent to carry out that approach. We shall now summarize our principal results.

In 2, we show that the free Cartesian monoid F is simple and has no non-trivial homomorphisms (it follows from this that the simply typed lambda calculus with surjective pairing is complete ([7])). Also in 2, we show that F is finitely generated (this is a key fact in the results that follow on unification). In 3, we consider the group G of F and the submonoid H of right invertible elements. We give simple syntactic characterizations of membership in G and H, and in addition, useful generators for G. In the process we locate the Freyd-Heller group as (anti isomorphic to) a subgroup of G. Also in 3,we observe that the wreath product of the number theoretic functions of finite support with F is embeddable in F. This construction is the basis for all of the recursion theoretic results proved below. In 4, we give a simple representation theorem for F in the spirit of the one for the Freyd-Heller group in [1]. The members of F are represented faithfully as the piecewise shift operators on Cantor space (again the representation is an anti isomorphism). The Freyd-Heller group turns up as the subgroup of order-preserving homeomorphisms. These observations are not used elsewhere but we think that they are of independent interest. In 5, we generalize many of the above results to the polynomial monoid F [x].In particular, we show that it is separable. Moving to F [x] permits the generalization of the results for F to all higher types in the lambda calculus (via the theory of reducibility; see


Normal Form Match Problem Wreath Product Invertible Element Lambda Calculus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Barendregt, “The Lambda Calculus”, North Holland, 1981.Google Scholar
  2. [2]
    Freyd & Heller, Splitting homotopy invariants (unpublished manuscript to appear in Festschrift for Alex Heller).Google Scholar
  3. [3]
    Klop, Combinatory Reduction Systems, Mathematical Centre Tracts, 127, Math Centrum Amsterdam, 1980.Google Scholar
  4. [4]
    Lambek, From lambda calculus to Cartesian closed categories, Seldin & Hindley (eds.), To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Academic Press, 1980 (Curry festschrift).Google Scholar
  5. [5]
    Matiyasevich, Diophantine representation of recursively enumerable predicates in Fenstad ed., Proceedings of the Second Scandanavian Logic Symposium. North Holland, 1971.Google Scholar
  6. [6]
    Scott, Relating theories of the lambda calculus, Curry Festschrift.Google Scholar
  7. [7]
    Statman, Freyd's hierarchy of combinator monoids in Proceedings Symposium on Logic in Computer Science, IEEE, 1991.Google Scholar
  8. [8]
    Statman, Simply typed Lambda Calculus with Surjective Pairing, CMU, Dept. of Mathematical Sciences, Report 92-146.Google Scholar
  9. [9]
    Statman, Recursive types and the subject reduction theorem, CMU, Dept. of Mathematical Sciences, Report 94-164.Google Scholar
  10. [10]
    Statman, A local translation of untyped lambda calculus into simply typed lambda calculus, CMU, Dept. of Mathematical Sciences, Report 91-134.Google Scholar
  11. [11]
    Statman, Combinators and the theory of partitions, CMU, Dept. of Mathemtical Sciences, Report 88-31.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Rick Statman
    • 1
  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburgh

Personalised recommendations