Factorization in Topological Monoids

We sketch a theory of divisibility and factorization in topological monoids, where finite products are replaced by convergent products. The algebraic case can then be viewed as the special case of discretely topologized topological monoids. We define the topological factorization monoid, a generalization of the factorization monoid for algebraic monoids, and show that it is always topologically factorial : any element can be uniquely written as a convergent product of irreducible elements. We give some examples of topologically factorial monoids. Furthermore, we investigate some topological properties related to the convergence of infinite products. 1. A primer on algebraic monoids In this section, we give some basic definitions on the divisibility and factorization theory of algebraic monoids. For additional information we refer to [6, 5] 1.1. Divisibility. An (algebraic) monoid H is a semi-group with a neutral element. In this paper, H is assumed to be abelian and cancellative. Unless otherwise stated, we write H multiplicatively and denote by 1 ∈ H its neutral element. Only the monoid N = (N,+) will be written additively, with 0 ∈ N the neutral element. We denote the set of units in H by H, and say that H is reduced if H = {1}. Since H is a subgroup of H , we can form the factor monoid Hred = H/H , which is reduced. More explicitly, Hred = H/ ∼ where a ∼ b iff a = eb for some e ∈ H. If a, b ∈ H then we say that a divides b, and write a |b , if there exists a (necessarily unique) element c ∈ H such that ac = b. An element p ∈ H \H is said said to be irreducible if p = ab with a, b ∈ H implies that either a or b is a unit. The irreducible elements in H are called atoms; we denote by A(H) the set of all atoms in H . We say that p ∈ H \H is prime if whenever p divides ab, it divides either a or b. Note that a prime element is always irreducible. For a set P , we denote by F(P ) the free abelian monoid on P . If M ⊂ H is a subset of H , we denote by [M ] the submonoid generated by M . 1.2. Factorization. Let H be an abelian, cancellative and reduced algebraic monoid. The free abelian monoid F(A(H)) is called the factorization monoid of H , and the canonical homomorphism πH : F(A(H)) → H (1) is called the factorization homomorphism. If p ∈ H , then the elements in π H (p) are called the factorizations of p. Definition 1.1. A monoid H is atomic if the following equivalent conditions are fulfilled: i) Every p ∈ H \H can be written as a finite product of atoms, ii) H = [A(H)], iii) The factorization homomorphism πH is surjective. Definition 1.2. A monoid H is called factorial if the following equivalent conditions are fulfilled: Date: March 9, 2008. 1991 Mathematics Subject Classification. 20M14, 22.05 (Primary) 21.0X (Secondary).