Editing Idempotence (section)

== Definition ==
An element <math>x</math> of a set <math>S</math> equipped with a [[binary operator]] <math>\cdot</math> is said to be ''idempotent'' under <math>\cdot</math> if<ref>{{cite book |last=Valenza |first=Robert |date=2012 |title=Linear Algebra: An Introduction to Abstract Mathematics |url=https://books.google.com/books?id=7x8MCAAAQBAJ |location=Berlin |publisher=Springer Science & Business Media |page=22 |isbn=9781461209010 |quote=An element ''s'' of a magma such that ''ss'' = ''s'' is called ''idempotent''.}}</ref><ref>{{cite book |last=Doneddu |first=Alfred |date=1976 |title=Polynômes et algèbre linéaire |url=https://books.google.com/books?id=5Ry7AAAAIAAJ |language=fr |location=Paris |publisher=Vuibert |page=180 |quote=Soit ''M'' un magma, noté multiplicativement. On nomme idempotent de ''M'' tout élément ''a'' de ''M'' tel que ''a''<sup>2</sup> = ''a''.}}</ref>
: {{nowrap|1=<math>x\cdot x=x</math>}}.
The ''binary operation'' <math>\cdot</math> is said to be ''idempotent'' if<ref>{{cite book | author=George Grätzer | title=General Lattice Theory | url=https://archive.org/details/generallatticeth0000grat | url-access=registration | location=Basel | publisher=Birkhäuser | year=2003 | isbn=978-3-7643-6996-5 }} Here: Sect.1.2, p.5.</ref><ref>{{cite book | author=Garrett Birkhoff | title=Lattice Theory | location=Providence | publisher=Am. Math. Soc. | series=Colloquium Publications | volume=25 | year=1967 }}. Here: Sect.I.5, p.8.</ref>
: {{nowrap|1=<math>x\cdot x=x</math> for all <math>x\in S</math>}}.