A related, but weaker notion than the existence of normal forms is that of two objects being joinable: x and y are said joinable if there exists some z with the property that x \stackrel{*}{\rightarrow} z \stackrel{*}{\leftarrow} y.
An ARS is said to possess the Church-Rosser property if and only if x \stackrel{*}{\leftrightarrow} y implies x\mathbin\downarrow y for all objects x, y.