# Simple semi-group

A 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) $0$- 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 ${\mathcal D}$- class (cf. Green equivalence relations); $0$- bi-simple semi-groups, consisting of two ${\mathcal D}$- 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 ${\mathcal D}$- classes consist of one element). The most important type of ideal-simple semi-groups ( $0$- simple semi-groups) are the completely-simple semi-groups (completely $0$- simple semi-groups, cf. 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 $\mathop{\rm Sp} _ {4}$( cf. Bicyclic semi-group; ). The latter, $\mathop{\rm Sp} _ {4}$, is given by generators $a , b , c , d$ and defining relations $a ^ {2} = a$, $b ^ {2} = b$, $c ^ {2} = c$, $d ^ {2} = d$, $b a = a$, $a b = b$, $b c = b$, $c b = c$, $d c = c$, $c d = d$, $d a = d$. It is isomorphic to a Rees semi-group of matrix type over a bicyclic semi-group with generators $u , v$, where $u v = 1$, with sandwich-matrix

$$\left \| \begin{array}{cc} 1 & v \\ 1 & 1 \\ \end{array} \right \| .$$

In a sense, $\mathop{\rm Sp} _ {4}$ 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 $a , b$ there is an $x$ such that $a x = b$. The right simple semi-groups containing idempotents are precisely the right groups (cf. Right group). An important example of a right simple semi-group without idempotents is given by the semi-groups $T ( M , \delta , p , q )$ of all transformations $\phi$ of a set $M$ such that: 1) the kernel of $\phi$ is the equivalence relation $\delta$ on $M$; 2) the cardinality of the quotient set $M / \delta$ is $p$; 3) the set $M \phi$ intersects each $\delta$- class in at most one element; and 4) the set of $\delta$- classes disjoint from $M \phi$ has infinite cardinality $q$, and $q \leq p$. The semi-group $T ( M , \delta , p , q )$ is called a Teissier semi-group of type $( p , q )$, and, if $\delta$ is the equality relation, it is called a Baer–Levi semi-group of type $( p , q )$( cf. , ). 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 $p = q$).

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; Brandt semi-group; Right group; for bi-simple inverse semi-groups (including structure theorems under certain restrictions on the semi-lattice of idempotents) see , , . There are ideal-simple inverse semi-groups with an arbitrary number of ${\mathcal D}$- 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. ), in a bi-simple semi-group generated by idempotents (cf. ), 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. ).

How to Cite This Entry:
Simple semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simple_semi-group&oldid=49585
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article