Magma

2020 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" , Springer (1958)
[5]	N. Bourbaki, "Algebra", 1, Chap.1-3, Springer (1989)