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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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{{TEX|done}}{{MSC|08A}}
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''groupoid''
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A [[universal algebra]] with one [[binary operation]]: a set $M$ endowed with an everywhere defined $m : M \times M \rightarrow M$ on it. No conditions are imposed. In particular, a magma need not be [[Commutativity|commutative]] or [[Associativity|associative]]:  it is the broadest class of such algebras: groups, semi-groups, quasi-groups – all these are magmas of a special type. .
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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.
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An important concept in the theory of magma is that of isotopy of operations. On a set $G$ let there be defined two binary operations, denoted by $(\cdot)$ and $(\circ)$; they are isotopic if there exist three one-to-one mappings $\alpha$, $\beta$ and $\gamma$ of $G$ onto itself such that $a\cdot b=\gamma^{-1}(\alpha a\circ\beta b)$ for all $a,b\in G$ (cf. [[Isotopy (in algebra)]]). A magma that is isotopic to a [[Quasi-group|quasi-group]] is itself a quasi-group; a magma with a unit element that is isotopic to a group, is also isomorphic to this group. For this reason, in group theory the concept of isotopy is not used: For groups isotopy and isomorphism coincide.
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A magma with cancellation is a magma in which either of the equations $ab=ac$, $ba=ca$ implies $b=c$, where $a$, $b$ and $c$ are elements of the magma. Any magma with cancellation is imbeddable into a quasi-group. A homomorphic image of a quasi-group is a magma with division, that is, a magma in which the equations $ax=b$ and $ya=b$ are solvable (but do not necessarily have unique solutions).
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Of particular importance is the [[free magma]] on an alphabet (set) $X$.
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A set with one partial binary operation (i.e. one not defined for all pairs of elements) is said to be a partial magma. Any partial  submagma of a free partial magma is free.
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====References====
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  O. Boruvka,  "Foundations of the theory of groupoids and groups" , Wiley  (1976)  (Translated from German)</TD></TR>
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<TR><TD valign="top">[4]</TD> <TD valign="top">  R.H. Bruck,  "A survey of binary systems" Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge. '''20''' Springer  (1958) {{ZBL|0081.01704}}</TD></TR>
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<TR><TD valign="top">[5]</TD> <TD valign="top">  N. Bourbaki, "Algebra", '''1''', Chap.1-3, Springer (1989) </TD></TR>
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</table>

Latest revision as of 21:12, 7 January 2016

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

groupoid

A universal algebra with one binary operation: a set $M$ endowed with an everywhere defined $m : M \times M \rightarrow M$ on it. No conditions are imposed. In particular, a magma need not be commutative or associative: it is the broadest class of such algebras: groups, semi-groups, quasi-groups – all these are magmas of a special type. .

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.

An important concept in the theory of magma is that of isotopy of operations. On a set $G$ let there be defined two binary operations, denoted by $(\cdot)$ and $(\circ)$; they are isotopic if there exist three one-to-one mappings $\alpha$, $\beta$ and $\gamma$ of $G$ onto itself such that $a\cdot b=\gamma^{-1}(\alpha a\circ\beta b)$ for all $a,b\in G$ (cf. Isotopy (in algebra)). A magma that is isotopic to a quasi-group is itself a quasi-group; a magma with a unit element that is isotopic to a group, is also isomorphic to this group. For this reason, in group theory the concept of isotopy is not used: For groups isotopy and isomorphism coincide.

A magma with cancellation is a magma in which either of the equations $ab=ac$, $ba=ca$ implies $b=c$, where $a$, $b$ and $c$ are elements of the magma. Any magma with cancellation is imbeddable into a quasi-group. A homomorphic image of a quasi-group is a magma with division, that is, a magma in which the equations $ax=b$ and $ya=b$ are solvable (but do not necessarily have unique solutions).

Of particular importance is the free magma on an alphabet (set) $X$.

A set with one partial binary operation (i.e. one not defined for all pairs of elements) is said to be a partial magma. Any partial submagma of a free partial magma is free.

References

[1] A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)
[2] P.M. Cohn, "Universal algebra" , Reidel (1981)
[3] O. Boruvka, "Foundations of the theory of groupoids and groups" , Wiley (1976) (Translated from German)
[4] R.H. Bruck, "A survey of binary systems" Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge. 20 Springer (1958) Zbl 0081.01704
[5] N. Bourbaki, "Algebra", 1, Chap.1-3, Springer (1989)
How to Cite This Entry:
Magma. Encyclopedia of Mathematics. URL: http://www.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