Call a binary relationMathworldPlanetmath on a set S a reductionPlanetmathPlanetmath. Let * be the reflexive transitive closure of . Two elements a,bS are said to be joinable if there is an element cS such that a*c and b*c. Graphically, this means that


where the starred arrows represent

aa1anc   and   bb1bmc

respectively (m,n are non-negative integers). The case m=0 (or n=0) means ac (or bc).

Definition. is said to be confluent if whenever x*a and x*b, then a,b are joinable. In other words, is confluent iff * has the amalgamation property. Graphically, this says that, whenever we have a diagram


then it can be “completed” into a “diamond”:


Remark. A more general property than confluence, called semi-confluence is defined as follows: is semi-confluent if for any triple x,a,bS such that xa and x*b, then a,b are joinable. It turns out that this seemingly weaker notion is actually equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the stronger notion of confluence. In additionPlanetmathPlanetmath, it can be shown that is confluent iff has the Church-Rosser propertyMathworldPlanetmath.


  • 1 F. Baader, T. Nipkow, Term Rewriting and All That, Cambridge University Press (1998).
Title confluence
Canonical name Confluence
Date of creation 2013-03-22 17:47:24
Last modified on 2013-03-22 17:47:24
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 14
Author CWoo (3771)
Entry type Definition
Classification msc 68Q42
Related topic ChurchRosserProperty
Related topic AmalgamationProperty
Related topic NormalizingReduction
Related topic TerminatingReduction
Defines confluent
Defines joinable
Defines semi-confluent