Partial monoids: associativity and confluence

A partial monoid P is a set with a partial multiplication × (and total identity 1P ) which satisfies some associativity axiom. The partial monoid P may be embedded in the free monoid P ∗ and the product × is simulated by a string rewriting system on P ∗ that consists in evaluating the concatenation of two letters as a product in P , when it is defined, and a letter 1P as the empty word ǫ. In this paper we study the profound relations between confluence for such a system and associativity of the multiplication. Moreover we develop a reduction strategy to ensure confluence and which allows us to define a multiplication on normal forms associative up to a given congruence of P ∗. Finally we show that this operation is associative if, and only if, the rewriting system under consideration is confluent. Key-words: Partial monoid, string rewriting system, normal form, associativity and confluence