Unique factorization monoids and domains

It is the purpose of this paper to construct unique factorization (uf) monoids and domains. The principal results are: (1) The free product of a well-ordered set of monoids is a uf-monoid iff every monoid in the set is a uf-monoid. (2) If M is an ordered monoid and F is a field, the ring F [AI] ] of all formal power series with well-ordered support is a uf-domain iff M is naturally ordered (i.e., whenever b 1 }. An ordered monoid M is said to be naturally ordered (see [4, p. 154]) iff whenever aMnbM$0 and b<a, then aMCbM. It is the purpose of this paper to show ways of constructing ufmonoids and uf-domains. The two principal results are as follows. THEOREM 1. The free product of a well-ordered set of monoids is a uf-monoid if every monoid in the set is a uf-monoid. THEOREM 2. Let M be an ordered monoid and F be a field. The ring F[[M]] of all formal power series with well-ordered support is a ufdomain if M is naturally ordered. Received by the editors April 27, 1970. AMS 1970 subject classifications. Primary 16A02, 20M25; Secondary 06A50.