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; [11]). 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. [6], [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 $ 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 [1], [8], [9]. 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. [1]), in a bi-simple semi-group generated by idempotents (cf. [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. [3]–[5]).

References

[1]	A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)
[2]	E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian)
[3]	L.A. Bokut', "Some embedding theorems for rings and semigroups" Sibirsk. Mat. Zh. , 4 : 3 (1963) pp. 500–518 (In Russian)
[4]	E.G. Shutov, "Embeddings of semigroups in simple and complete semigroups" Mat. Sb. , 62 : 4 (1963) pp. 496–511 (In Russian)
[5]	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
[6]	R. Baer, F. Levi, "Vollständige irreduzibele Systeme von Gruppenaxiomen" Sitzungsber. Heidelb. Akad. Wissenschaft. Math.-Nat. Kl. , 2 (1932) pp. 3–12
[7]	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
[8]	W.D. Munn, "Some recent results on the structure of inverse semigroups" K.W. Folley (ed.) , Semigroups , Acad. Press (1969) pp. 107–123
[9]	J.M. Howie, "An introduction to semigroup theory" , Acad. Press (1976)
[10]	F. Pastijn, "Embedding semigroups in semibands" Semigroup Forum , 14 : 3 (1977) pp. 247–263
[11]	K. Byleen, J. Meakin, F. Pastijn, "The fundamental four-spiral semigroup" J. of Algebra , 54 (1978) pp. 6–26