On Properties of Monoids That Are Modular for Free Products and for Certain Free Products with Amalgamated Submonoids

A property P of string-rewriting systems is called modular if the disjoint union R 1 R 2 of two string-rewriting systems R 1 and R 2 has this property if and only if R 1 and R 2 both have this property. Analogously, a property P of monoids is modular if the free product M 1 M 2 of two monoids M 1 and M 2 has this property if and only if M 1 and M 2 both have this property. Since the string-rewriting systems form a subclass of the linear term-rewriting systems, it follows that many properties are modular for string-rewriting systems. Here we give a summary of these modularity results, providing fairly simple proofs for them. In addition, we show that the property of having nite derivation type (FDT) is modular for nitely presented monoids. In a second part we consider the algebraic operation of forming a free product with amalgamating certain submonoids, which corresponds to a non-disjoint union of string-rewriting systems. Following Toyama and Aoto (1996) we show that certain properties like termination and connuence are modular also for this type of combining systems. In particular, we prove that the property FDT remains modular in this setting.