A semi-group with a zero element in which to each non-zero element $a$ there correspond uniquely determined elements $e,f,a'\in S$ such that $ea=af=a$ and $a'a=f$, and for any two non-zero idempotents $g_1,g_2\in S$ one has $g_1Sg_2\neq0$. The elements $e$ and $f$ in the definition will in fact be idempotent and $fa'=a'e=a'$ and $aa'=e$. Moreover, in a Brandt semi-group each one of the conditions $ac=bc\neq0$, $ca=cb\neq0$, implies $a=b$, while the conditions $ab\neq0$ and $bc\neq0$ imply $abc\neq0$.
The partial groupoid obtained by excluding the zero element from a Brandt semi-group is said to be a Brandt groupoid. The concept of a Brandt semi-group was introduced by H. Brandt in , and the concept of a Brandt groupoid was introduced in the same publication by implication. The concept of a Brandt groupoid is an abstraction of the system of normal ideals of semi-simple linear algebras with respect to the so-called proper multiplication (cf. , , Chapt. 6). The significance of a Brandt semi-group in the theory of semi-groups lies in the fact that Brandt semi-groups are just completely $0$-simple inversion semi-groups (cf. Inversion semi-group; Completely-simple semi-group). A semi-group will be a Brandt semi-group if and only if it is isomorphic to a Rees semi-group of matrix type with a unit sandwich matrix over a group with an added zero element.
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Brandt semi-group. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Brandt_semi-group&oldid=31810