View source for Simple semi-group

A [[Semi-group|semi-group]] not containing proper ideals or congruences of some fixed type. Various kinds of simple semi-groups arise, depending on the type considered: ideal-simple semi-groups, not containing proper two-sided ideals (the term simple semi-group is often used for such semi-groups only); left (right) simple semi-groups, not containing proper left (right) ideals; (left, right) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s0853002.png" />-simple semi-groups, semi-groups with a zero not containing proper non-zero two-sided (left, right) ideals and not being two-element semi-groups with zero multiplication; bi-simple semi-groups, consisting of one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s0853003.png" />-class (cf. [[Green equivalence relations|Green equivalence relations]]); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s0853005.png" />-bi-simple semi-groups, consisting of two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s0853006.png" />-classes one of which is the null class; and congruence-free semi-groups, not having congruences other than the universal relation and the equality relation.

Every left or right simple semi-group is bi-simple; every bi-simple semi-group is ideal-simple, but there are ideal-simple semi-groups that are not bi-simple (and even ones for which all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s0853007.png" />-classes consist of one element). The most important type of ideal-simple semi-groups (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s0853008.png" />-simple semi-groups) are the completely-simple semi-groups (completely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s0853009.png" />-simple semi-groups, cf. [[Completely-simple semi-group|Completely-simple semi-group]]). The most important examples of bi-simple but not completely-simple semi-groups are: the bicyclic semi-groups and the four-spiral semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530010.png" /> (cf. [[Bicyclic semi-group|Bicyclic semi-group]]; [[#References|[11]]]). The latter, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530011.png" />, is given by generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530012.png" /> and defining relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530023.png" />. It is isomorphic to a [[Rees semi-group of matrix type|Rees semi-group of matrix type]] over a bicyclic semi-group with generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530025.png" />, with sandwich-matrix

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530026.png" /></td> </tr></table>

In a sense, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530027.png" /> is minimal among the bi-simple not completely-simple semi-groups generated by a finite number of idempotents, and quite often it arises as a sub-semi-group of those semi-groups.

Right simple semi-groups are also called semi-groups with right division, or semi-groups with right invertibility. The reason for this terminology is the following equivalent property of such semi-groups: For any elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530028.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530029.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530030.png" />. The right simple semi-groups containing idempotents are precisely the right groups (cf. [[Right group|Right group]]). An important example of a right simple semi-group without idempotents is given by the semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530031.png" /> of all transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530032.png" /> of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530033.png" /> such that: 1) the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530034.png" /> is the equivalence relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530035.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530036.png" />; 2) the cardinality of the quotient set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530037.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530038.png" />; 3) the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530039.png" /> intersects each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530040.png" />-class in at most one element; and 4) the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530041.png" />-classes disjoint from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530042.png" /> has infinite cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530043.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530044.png" />. The semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530045.png" /> is called a Teissier semi-group of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530046.png" />, and, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530047.png" /> is the equality relation, it is called a Baer–Levi semi-group of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530048.png" /> (cf. [[#References|[6]]], [[#References|[7]]]). A Teissier semi-group is an example of a right simple semi-group without idempotents that does not necessarily satisfy the right cancellation law. Every right simple semi-group without idempotents can be imbedded in a suitable Teissier semi-group, while every such semi-group with the right cancellation law can be imbedded in a suitable Baer–Levi semi-group (in both cases one can take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530049.png" />).

Various types of simple semi-groups often arise as  "blocks"  from which one can construct the semi-groups under consideration. For classical examples of simple semi-groups see [[Completely-simple semi-group|Completely-simple semi-group]]; [[Brandt semi-group|Brandt semi-group]]; [[Right group|Right group]]; for bi-simple inverse semi-groups (including structure theorems under certain restrictions on the semi-lattice of idempotents) see [[#References|[1]]], [[#References|[8]]], [[#References|[9]]]. There are ideal-simple inverse semi-groups with an arbitrary number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085300/s08530050.png" />-classes. In the study of imbedding of semi-groups in simple semi-groups one usually either indicates conditions for the possibility of the corresponding imbedding, or establishes that any semi-group can be imbedded in a semi-group of the type considered. E.g., any semi-group can be imbedded in a bi-simple semi-group with an identity (cf. [[#References|[1]]]), in a bi-simple semi-group generated by idempotents (cf. [[#References|[10]]]), and in a semi-group that is simple relative to congruences (which may have some property given in advance: the presence or absence of a zero, completeness, having an empty Frattini sub-semi-group, etc., cf. [[#References|[3]]]–[[#References|[5]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.H. Clifford,   G.B. Preston,   "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc.  (1961–1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.S. Lyapin,   "Semigroups" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.A. Bokut',   "Some embedding theorems for rings and semigroups"  ''Sibirsk. Mat. Zh.'' , '''4''' :  3  (1963)  pp. 500–518  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.G. Shutov,   "Embeddings of semigroups in simple and complete semigroups"  ''Mat. Sb.'' , '''62''' :  4  (1963)  pp. 496–511  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.N. Klimov,   "Embedding of semigroups in factorizable semigroups"  ''Sib. Math. J.'' , '''14''' :  5  (1973)  pp. 715–722  ''Sibirsk. Mat. Zh.'' , '''14''' :  5  (1973)  pp. 1025–1036</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R. Baer,   F. Levi,   "Vollständige irreduzibele Systeme von Gruppenaxiomen"  ''Sitzungsber. Heidelb. Akad. Wissenschaft. Math.-Nat. Kl.'' , '''2'''  (1932)  pp. 3–12</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  M. Teissier,   "Sur les demi-groupes admettant l'existence du quotient d'un cote"  ''C.R. Acad. Sci. Paris'' , '''236''' :  11  (1953)  pp. 1120–1122</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  W.D. Munn,   "Some recent results on the structure of inverse semigroups"  K.W. Folley (ed.) , ''Semigroups'' , Acad. Press  (1969)  pp. 107–123</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  J.M. Howie,   "An introduction to semigroup theory" , Acad. Press  (1976)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  F. Pastijn,   "Embedding semigroups in semibands"  ''Semigroup Forum'' , '''14''' :  3  (1977)  pp. 247–263</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  K. Byleen,   J. Meakin,   F. Pastijn,   "The fundamental four-spiral semigroup"  ''J. of Algebra'' , '''54'''  (1978)  pp. 6–26</TD></TR></table>