A diamond is a finite directed graph without oriented cycles which is a union of two (directed) chains with common minimum and common maximum (some other intermediate points and even edges may be in common as well).
It is often interesting wheather a given span in some partial ordered set can be completed into a diamond. The property of a collection of spans to consist of spans which are expandable into diamonds is very useful in the theory of rewriting systems and producing normal forms in algebra. There are classical results e.g. Newman’s diamond lemma, Širšov-Bergman’s diamond lemma (Širšov is also sometimes spelled as Shirshov), and Church-Rosser theorem? (and the corresponding Church-Rosser confluence property).
L. A. Bokutʹ, I. P. Shestakov, Some results by A. I. Shirshov and his school, Second International Conference on Algebra (Barnaul, 1991), 1–12, Contemp. Math. 184, Amer. Math. Soc. 1995.
L. A. Bokut’, Unsolvability of the word problem and subalgebras of finitely presented Lie algebras, Izv. Akad. Nauk S.S.S.R. Ser. Mat. 36 (1972), 1173-1219 (Russian).
Distinguish from the 5-edge 4-point undirected diamond graph.