Binary operation on magma determines neutral element: Difference between revisions

Latest revision as of 03:52, 6 July 2019

Statement

Suppose $(S, *)$ is a magma (set $S$ with binary operation $*$ ). Then, if there exists a neutral element for $*$ (i.e., an element $e$ such that $e * a = a * e = a$ for all $a \in S$ ), the element $e$ is uniquely determined by $*$ .

In other words, a magma can have at most one two-sided neutral element.

Facts used

Equality of left and right neutral element

Related facts

In the case that $*$ is associative, this says that the identity element (neutral element) of a monoid is completely determined by the binary operation. This yields the fact that monoids form a non-full subcategory of semigroups.

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 ==Statement==
-Suppose <math>(S,*)</math> is a [[magma]] (set <math>S</math> with associative binary operation <math>*</math>). Then, if there exists a [[neutral element]] for <math>*</math> (i.e., an element <math>e</math> such that <math>e * a = a * e = a</math> for all <math>a \in S</math>), the element <math>e</math> is uniquely determined by <math>*</math>.
+Suppose <math>(S,*)</math> is a [[magma]] (set <math>S</math> with binary operation <math>*</math>). Then, if there exists a [[neutral element]] for <math>*</math> (i.e., an element <math>e</math> such that <math>e * a = a * e = a</math> for all <math>a \in S</math>), the element <math>e</math> is uniquely determined by <math>*</math>.
 In other words, a magma can have at most one two-sided neutral element.