# On Cartesian monoids

## Abstract

A Cartesian monoid is a structure (*M*, *, *I, L, R*, 〈<>〉) where (*M, *, I*) is a monoid with *L* and *R* members of *M* and 〈〉: *M*^{2} → *M* s.t. *L* * 〈*x, y*〉=*x R* * 〈*x, y*〉=*x* 〈*x, 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

## Keywords

Normal Form Match Problem Wreath Product Invertible Element Lambda Calculus## Preview

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## References

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