Transformation semi-group

Any sub-semi-group of a symmetric semi-group $T_\Omega$, where $T_\Omega$ is the set of all transformations of a set $\Omega$. Particular cases of transformation semi-groups are transformation groups (cf. Transformation group).

Two transformation semi-groups $P_1 \subset T_{\Omega_1}$, $P_2 \subset T_{\Omega_2}$ are called similar if there are bijections $\phi : \Omega_1 \rightarrow \Omega_2$ and $\psi : P_1 \rightarrow P_2$ such that $u\alpha = \beta$ ($\alpha, \beta \in \Omega_1$, $u \in P_1$) implies $(\psi u) (\phi\alpha) = \phi\beta$. Similar transformation semi-groups are isomorphic, but the converse is, usually, not true. However, within some classes of transformation semi-groups isomorphism implies similarity. E.g., the class of transformation semi-groups that include all transformations $u$ such that $u\Omega$ consists of one element. The specification of a semi-group as a transformation semi-group includes more information than its specification up to isomorphism.

Distinguishing properties of transformation semi-groups that are invariant under isomorphism is of prime importance. For a given class $\Gamma$ of transformation semi-groups, conditions under which a semi-group $S$ is isomorphic to some semi-group from $\Gamma$ are called abstract characteristics of the class $\Gamma$. Abstract characteristics for certain important classes of transformation semi-groups have been found. Every semi-group is isomorphic to some transformation semi-group. A semi-group $S$ is isomorphic to some symmetric semi-group $T_\Omega$ if it is a maximal complete ideal extension (cf. Extension of a semi-group) of any semi-group $A$ with the identity $xy=x$.

One distinguishes directions in the general theory of transformation semi-groups in which the set $\omega$ to be transformed is endowed with a certain structure (a topology, an action, a relation in $\Omega$, etc.) and considers transformation semi-groups related to this structure (endomorphisms, continuous or linear transformations, translations of semi-groups, etc.). The study of relations between properties of the structure in $\Omega$ and properties of the semi-groups of corresponding transformations is a generalization of Galois theory. In particular, cases are known in which the indicated translation semi-group completely determines the structure (cf. e.g. Endomorphism semi-group). Properties of left and right translations of semi-groups are used in general semi-group theory.

A generalization of the notion of a transformation is that of a partial transformation, mapping some subset $\Omega' \subset \Omega$ into $\Omega$. Binary relations on a set $\Omega$ are sometimes treated as multi-valued (in general, partial) transformations of this set. Single- and multi-valued partial transformations also form semi-groups under the operation of composition (regarded as multiplication of binary relations). It is expedient to regard them as semi-groups endowed with additional structures (e.g. the relation of inclusion of binary relations, inclusion or equality of domains of definition, inclusion or equality of ranges, etc.).

References

[1]	E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian)
[2]	A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)
[3]	L.M. Gluskin, "Ideals of semigroups" Mat. Sb. , 55 : 4 (1961) pp. 421–428 (In Russian)
[4]	B.M. Schein, "Relation algebras and function semigroups" Semigroup Forum , 1 : 1 (1970) pp. 1–62