Brandt semi-group

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 [1], 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. [2], [3], 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.

References

[1]	H. Brandt, "Ueber eine Verallgemeinerung des Gruppenbegriffes" Math. Ann. , 96 (1927) pp. 360–366
[2]	M. Deuring, "Algebren" , Springer (1935)
[3]	N. Jacobson, "The theory of rings" , Amer. Math. Soc. (1943)
[4]	A.K. Sushkevich, "The theory of generalized groups" , Khar'kov-Kiev (1937) (In Russian)
[5]	A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)