Binary operation on magma determines neutral element: Difference between revisions
(New page: ==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>*</mat...) |
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==Statement== | ==Statement== | ||
Suppose <math>(S,*)</math> is a [[magma]] (set <math>S</math> with | 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. | In other words, a magma can have at most one two-sided neutral element. | ||
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[[Equality of left and right neutral element]] | [[Equality of left and right neutral element]] | ||
==Related facts== | |||
In the case that <math>*</math> 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]]. | |||
Latest revision as of 03:52, 6 July 2019
Statement
Suppose is a magma (set with binary operation ). Then, if there exists a neutral element for (i.e., an element such that for all ), the element 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.