# 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.).

How to Cite This Entry:
Transformation semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transformation_semi-group&oldid=36985
This article was adapted from an original article by L.M. GluskinE.S. Lyapin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article