### Partial monoids: associativity and confluencePartial monoids: associativity and confluence

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 Author Poinsot, Laurent ♦ Duchamp, Gérard ♦ Tollu, Christophe Source arXiv.org Content type Text File Format PDF Date of Submission 2010-02-10 Language English
 Subject Domain (in DDC) Computer science, information & general works Subject Keyword Computer Science - Discrete Mathematics ♦ cs Abstract A partial monoid $P$ is a set with a partial multiplication $\times$ (and total identity $1_P$) which satisfies some associativity axiom. The partial monoid $P$ may be embedded in a free monoid $P^*$ and the product $\star$ 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 $1_P$ as the empty word $\epsilon$. 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. Description Reference: Journal of Pure and Applied Mathematics 3, 2 (2010) 265-285 Educational Use Research Learning Resource Type Article