Lemma (mathematics)

Not to be confused with Lemma (morphology).

In mathematics and other fields,^[a] a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem".^[3][4] In many cases, a lemma derives its importance from the theorem it aims to prove; however, a lemma can also turn out to be more important than originally thought.^[5]

Etymology

From the Ancient Greek λῆμμα, (perfect passive εἴλημμαι) something received or taken. Thus something taken for granted in an argument.^[6]

Comparison with theorem

There is no formal distinction between a lemma and a theorem, only one of intention (see Theorem terminology). However, a lemma can be considered a minor result whose sole purpose is to help prove a more substantial theorem – a step in the direction of proof.^[5]

Well-known lemmas

Some powerful results in mathematics are known as lemmas, first named for their originally minor purpose. These include, among others:

• Bézout's lemma
• Burnside's lemma
• Dehn's lemma
• Euclid's lemma
• Farkas' lemma
• Fatou's lemma
• Gauss's lemma (any of several named after Carl Friedrich Gauss)
• Greendlinger's lemma
• Itô's lemma
• Jordan's lemma
• Lovász local lemma
• Nakayama's lemma
• Poincaré's lemma
• Riesz's lemma
• Schur's lemma
• Schwarz's lemma
• Sperner's lemma
• Urysohn's lemma
• Vitali covering lemma
• Yoneda's lemma
• Zorn's lemma

While these results originally seemed too simple or too technical to warrant independent interest, they have eventually turned out to be central to the theories in which they occur.

See also

• Axiom
• Corollary
• Co-premise
• Fundamental lemma
• Inference objection
• List of lemmas
• Objection
• Porism
• Theorem
• Theorem terminology

Notes

1. Such as informal logic, argument mapping, and philosophy.^[1][2]