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Difference between revisions of "Magma"

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A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110040/m1100401.png" /> endowed with an everywhere defined [[Binary relation|binary relation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110040/m1100402.png" /> on it. No conditions are imposed. In particular, a magma need not be commutative or associative. Of particular importance is the [[Free magma|free magma]] on an alphabet (set) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110040/m1100403.png" />. A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110040/m1100404.png" /> of one magma into another is a morphism of magmas if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110040/m1100405.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110040/m1100406.png" />, i.e., if it respects the binary relations.
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A set $M$ endowed with an everywhere defined [[binary operation]] $m : M \times M \rightarrow M$ on it. No conditions are imposed. In particular, a magma need not be [[Commutativity|commutative]] or [[Associativity|associative]]. Of particular importance is the [[free magma]] on an alphabet (set) $X$. A mapping $f : N \rightarrow M$ of one magma into another is a morphism of magmas if $f(m_N(a,b)) = m_M(f(a),f(b))$ for all $a,b \in N$, i.e., if it respects the binary operations.

Revision as of 18:00, 21 November 2014


A set $M$ endowed with an everywhere defined binary operation $m : M \times M \rightarrow M$ on it. No conditions are imposed. In particular, a magma need not be commutative or associative. Of particular importance is the free magma on an alphabet (set) $X$. A mapping $f : N \rightarrow M$ of one magma into another is a morphism of magmas if $f(m_N(a,b)) = m_M(f(a),f(b))$ for all $a,b \in N$, i.e., if it respects the binary operations.

How to Cite This Entry:
Magma. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Magma&oldid=12223
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article