# Difference between revisions of "Magma"

(Importing text file) |
m (→References: expand bibliodata) |
||

(3 intermediate revisions by the same user not shown) | |||

Line 1: | Line 1: | ||

− | A | + | {{TEX|done}}{{MSC|08A}} |

+ | |||

+ | ''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 [[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. . | ||

+ | |||

+ | 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|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==== | ||

+ | <table> | ||

+ | <TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)</TD></TR> | ||

+ | <TR><TD valign="top">[2]</TD> <TD valign="top"> P.M. Cohn, "Universal algebra" , Reidel (1981)</TD></TR> | ||

+ | <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> | ||

+ | <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> | ||

+ | <TR><TD valign="top">[5]</TD> <TD valign="top"> N. Bourbaki, "Algebra", '''1''', Chap.1-3, Springer (1989) </TD></TR> | ||

+ | </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