Ivan Shishkin, Rye (1878)

Problems/General algebraExerciseReviewed

Unicity of the identity element

by Ancient Tree·
14
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English
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Let (M,∗)(M,*) be a unital magma with identity element ee. Show that ee is unique.

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Solution by Ancient Tree

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Suppose that e′e' is another identity element. In particular, this means that, for x:=ex:=e:
e∗e′=x∗e′=x=e.e*e'=x*e'=x=e .But since ee is an identity element, by definition, for y:=e′y:=e':
e∗e′=e∗y=y=e′.e*e'=e*y=y=e'.So we can conclude that e=e′e=e'.

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