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A semi-group with a zero element in which to each non-zero element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017600/b0176001.png" /> there correspond uniquely determined elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017600/b0176002.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017600/b0176003.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017600/b0176004.png" />, and for any two non-zero idempotents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017600/b0176005.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017600/b0176006.png" />. The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017600/b0176007.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017600/b0176008.png" /> in the definition will in fact be idempotent and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017600/b0176009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017600/b01760010.png" />. Moreover, in a Brandt semi-group each one of the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017600/b01760011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017600/b01760012.png" />, implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017600/b01760013.png" />, while the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017600/b01760014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017600/b01760015.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017600/b01760016.png" />.
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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 [[#References|[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. [[#References|[2]]], [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017600/b01760017.png" />-simple inversion semi-groups (cf. [[Inversion semi-group|Inversion semi-group]]; [[Completely-simple 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|Rees semi-group of matrix type]] with a unit sandwich matrix over a group with an added zero element.
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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 [[#References|[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. [[#References|[2]]], [[#References|[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|Inversion semi-group]]; [[Completely-simple 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|Rees semi-group of matrix type]] with a unit sandwich matrix over a group with an added zero element.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Brandt,  "Ueber eine Verallgemeinerung des Gruppenbegriffes"  ''Math. Ann.'' , '''96'''  (1927)  pp. 360–366</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Deuring,  "Algebren" , Springer  (1935)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Jacobson,  "The theory of rings" , Amer. Math. Soc.  (1943)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.K. Sushkevich,  "The theory of generalized groups" , Khar'kov-Kiev  (1937)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.H. Clifford,  G.B. Preston,  "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc.  (1961–1967)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Brandt,  "Ueber eine Verallgemeinerung des Gruppenbegriffes"  ''Math. Ann.'' , '''96'''  (1927)  pp. 360–366</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Deuring,  "Algebren" , Springer  (1935)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Jacobson,  "The theory of rings" , Amer. Math. Soc.  (1943)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.K. Sushkevich,  "The theory of generalized groups" , Khar'kov-Kiev  (1937)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.H. Clifford,  G.B. Preston,  "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc.  (1961–1967)</TD></TR></table>

Latest revision as of 10:45, 17 April 2014

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)
How to Cite This Entry:
Brandt semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brandt_semi-group&oldid=12423
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article