nLab
Brandt groupoid

History

H. Brandt axiomatically introduced in 1927 a class of partial binary algebraic structures and called them groupoids (German Gruppoid). Following Oystein Ore, all binary algebraic structures were soon called groupoids (now we say either binary algebraic structure or, following Bourbaki, a magma), hence Brandt groupoid is in general algebra often viewed as a class of partial groupoids. Contemporary notion of a connected groupoid is informationally equivalent to a Brandt groupoid. Hence Brandt groupoids in the new categorical format, and usually without the connectedness assumption, took over the name in mainstream mathematics, regarding the importance of the notion. Wikipedia simply now redirects Brandt groupoid to groupoid.

Remark

(usage of the terminology) In older literature, the specific class of groupoids, a codiscrete groupoid of a set XX is also sometimes called a Brandt groupoid (as mentioned in da Silva, Weinstein, Geometric models of noncommutative algebras).

Definition

A Brandt groupoid (M,)(M,\cdot) is a set with a partially defined binary operation \cdot such that

  1. (associativity) If aba\cdot b and bcb\cdot c are defined then (ab)c(a\cdot b)\cdot c and a(bc)a\cdot (b\cdot c) are defined and they are equal
  2. for each aMa\in M there are unique elements e,fMe,f\in M such that ea=af=ae\cdot a = a\cdot f = a, called respectively its left and right unit
  3. if the left units of aa and bb agree then there is xMx\in M such that ax=ba \cdot x = b; if the right units of cc and dd agree then there is yMy\in M such that yc=dy\cdot c = d
  4. (connectedness) if ee and ff are idempotents, then there is mm such that emfe\cdot m\cdot f is defined

Properties

The first three properties imply that the idempotents ee in MM are precisely the left units of all elements aa such that eae\cdot a is defined; they are also precisely the right units of all elements bb such that beb\cdot e is defined.

If (M,)(M,\cdot) is a Brandt groupoid then the set M{0}M\coprod \{0\} can be made into an inverse semigroup by extending \cdot so that the multiplication of any element with 00 is 00 and the product of any two elements in MM whose product was undefined is also 00. Semigroups of that kind are called Brandt semigroups.

Literature

  • H. Brandt, Über eine Verallgemeinerung des Gruppenbegriffes, Mathematische Annalen, (1927) 96 (1): 360–366, doi:10.1007/BF01209171
  • G. B. Preston, Congruences on Brandt semigroups, Mathematische Annalen 139:2 (1959) 91–94 doi
  • A. H. Clifford, Matrix representations of completely simple semigroups Amer. J. Math. 64, 327–342 (1942).

Last revised on February 2, 2018 at 07:55:02. See the history of this page for a list of all contributions to it.