# Difference between revisions of "Magma"

From Encyclopedia of Mathematics

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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. | 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 22:26, 21 November 2014

2010 Mathematics Subject Classification: *Primary:* 08A [MSN][ZBL]

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=34696

This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article