View source for Representation of a semi-group

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r0814101.png" /> in a class of semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r0814102.png" />''

A [[Homomorphism|homomorphism]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r0814103.png" /> into a semi-group from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r0814104.png" /> (in the case of an injective homomorphism one speaks of a faithful representation). One usually has in mind a class of concrete semi-groups. Representations in a class of transformation semi-groups (for short — representations by transformations, cf. also [[Transformation semi-group|Transformation semi-group]]), in a class of semi-groups of partial transformations or binary relations, or in a class of matrix semi-groups (so-called matrix, or linear, representations of a semi-group) have been studied most completely. In automata theory, with each [[Automaton|automaton]] one associates a representation of the free semi-group of transformations of the set of its internal states. Representations of a semi-group by transformations that are in some way or other related to properties of the elements of the transformed set, endowed with some structure, (endomorphisms, continuous transformations, etc.) have special significance. Every semi-group with an identity element can be isomorphically represented as the semi-group of all endomorphisms of an oriented or non-oriented graph, as the semi-group of all endomorphisms of an algebra with unary operations, etc. A few constructions for obtaining all representations of a semi-group by partial transformations are known (1983). These representations are constructed from certain simple representations by union, multiple repetition, restriction to a subset, and by the operation of [[Imbedding of semi-groups|imbedding of semi-groups]].

By adjoining an identity element to a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r0814105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r0814106.png" />, and by extending the [[Regular representation|regular representation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r0814107.png" /> by left shifts onto a Cartesian power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r0814108.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r0814109.png" />, one obtains a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r08141010.png" /> — the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r08141011.png" />-fold repetition of the regular representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r08141012.png" />. Every representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r08141013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r08141014.png" /> by transformations of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r08141015.png" /> can be obtained (cf. [[#References|[2]]]) from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r08141016.png" /> by means of a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r08141017.png" /> such that

<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/r/r081/r081410/r08141018.png" /></td> </tr></table>

Transitive representations (i.e. representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r08141019.png" /> by transformations of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r08141020.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r08141021.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r08141022.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r08141023.png" />) play a special role.

The representations of a semi-group are, by one-to-one partial transformations, related to the notion and properties of inverse semi-groups (cf. [[Inversion semi-group|Inversion semi-group]]).

For the study of matrix representations of a semi-group one invokes semi-group algebras (cf. [[Semi-group algebra|Semi-group algebra]]). The problem of reducibility of matrix representations has been studied. For a number of semi-groups (including finite semi-groups) the irreducible representations (cf. [[Irreducible representation|Irreducible representation]]) have been found. The matrix representations of completely-simple and completely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r08141024.png" />-simple semi-groups (cf. [[Completely-simple semi-group|Completely-simple semi-group]]) can be constructed as extensions of representations of their subgroups. The matrix representations of arbitrary semi-groups can be described by means of representations of their factors, which are simple and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081410/r08141025.png" />-simple semi-groups.

====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">  V.V. Vagner,   "Representations of ordered semi-groups"  ''Mat. Sb.'' , '''38''' :  2  (1956)  pp. 203–240  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.S. Lyapin,   "Representations of semi-groups by partial mappings"  ''Mat. Sb.'' , '''52''' :  1  (1960)  pp. 589–596  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B.M. Shain,   "Representations of semi-groups by binary relations"  ''Mat. Sb.'' , '''60''' :  3  (1963)  pp. 293–303  (In Russian)</TD></TR><TR><TD valign="top">[5a]</TD> <TD valign="top">  D.B. McAlister,   "Representations of semigroups by linear transformations I"  ''Semi-group Forum'' , '''2''' :  3  (1971)  pp. 189–263</TD></TR><TR><TD valign="top">[5b]</TD> <TD valign="top">  D.B. McAlister,   "Representations of semigroups by linear transformations II"  ''Semi-group Forum'' , '''2''' :  4  (1971)  pp. 283–320</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B. Jónsson,   "Topics in universal algebra" , Springer  (1972)</TD></TR></table>