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Magma (algebra)
In abstract algebra, a magma (or groupoid; not to be confused with groupoids in category theory) is a basic kind of algebraic structure. Specifically, a magma consists of a set, ''M'', equipped with a single binary operation, . The binary operation must be closed by definition but no other properties are imposed.
== History and terminology ==
The term ''groupoid'' was introduced in 1926 by Heinrich Brandt describing his Brandt groupoid (translated from the German ''Gruppoid''). The term was then appropriated by B. A. Hausmann and Øystein Ore (1937) in the sense (of a set with a binary operation) used in this article. In a couple of reviews of subsequent papers in Zentralblatt, Brandt strongly disagreed with this overloading of terminology. The Brandt groupoid is a groupoid in the sense used in category theory, but not in the sense used by Hausmann and Ore. Nevertheless, influential books in semigroup theory, including Clifford and Preston (1961) and Howie (1995) use groupoid in the sense of Hausmann and Ore. Hollings (2014) writes that the term ''groupoid'' is "perhaps most often used in modern mathematics" in the sense given to it in category theory.
According to Bergman and Hausknecht (1996): “There is no generally accepted word for a set with a not necessarily associative binary operation. The word ''groupoid'' is used by many universal algebraists, but workers in category theory and related areas object strongly to this usage because they use the same word to mean "category in which all morphisms are invertible". The term ''magma'' was used by Serre (Algebras and Lie Groups, 1965 ).” It also appears in Bourbaki's ''Éléments de mathématique'', Algèbre, chapitres 1 à 3, 1970.
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