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A semi-group consisting of the endomorphisms of a certain object (a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e0356201.png" /> endowed with some structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e0356202.png" />) with the operation of multiplication (performing transformations in succession). The object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e0356203.png" /> can be a vector space, a topological space, an algebraic system, a graph, etc.; it is usually regarded as an object of a certain [[Category|category]], and as a rule, the morphisms (cf. [[Morphism|Morphism]]) in this category are the mappings preserving the relations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e0356204.png" /> (linear or continuous transformations, homomorphisms, etc.). The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e0356205.png" /> of all endomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e0356206.png" /> (that is, of morphisms to its subobjects) is a sub-semi-group of the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e0356207.png" /> of all transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e0356208.png" /> (see [[Transformation semi-group|Transformation semi-group]]).
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A semi-group consisting of the endomorphisms of a certain object (a set $X$ endowed with some structure $\sigma$) with the operation of multiplication (performing transformations in succession). The object $X$ can be a vector space, a topological space, an algebraic system, a graph, etc.; it is usually regarded as an object of a certain [[Category|category]], and as a rule, the morphisms (cf. [[Morphism|Morphism]]) in this category are the mappings preserving the relations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e0356204.png" /> (linear or continuous transformations, homomorphisms, etc.). The set $\operatorname{End}X$ of all endomorphism of $X$ (that is, of morphisms to its subobjects) is a sub-semi-group of the semi-group $T_X$ of all transformations of $X$ (see [[Transformation semi-group|Transformation semi-group]]).
  
The semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e0356209.png" /> may include a considerable amount of information on the structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e03562010.png" />. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e03562011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e03562012.png" /> are vector spaces of dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e03562013.png" /> over skew-fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e03562014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e03562015.png" />, respectively, then if the semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e03562016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e03562017.png" /> of their endomorphisms (that is, linear transformations) are isomorphic, it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e03562018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e03562019.png" /> (and in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e03562020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e03562021.png" />) are isomorphic. Some pre-ordered sets and lattices, every Boolean ring, and some other algebraic systems are determined up to isomorphism by their endomorphism semi-groups. The same is true for some modules and transformation semi-groups. Similar information about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e03562022.png" /> is carried by certain proper sub-semi-groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e03562023.png" /> (for example, the semi-groups of homeomorphic transformations of a topological space).
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The semi-group $\operatorname{End}X$ may include a considerable amount of information on the structure $\sigma$. For example, if $X$ and $Y$ are vector spaces of dimensions $\geq2$ over skew-fields $F$ and $H$, respectively, then if the semi-groups $\operatorname{End}X$ and $\operatorname{End}Y$ of their endomorphisms (that is, linear transformations) are isomorphic, it follows that $X$ and $Y$ (and in particular, $F$ and $H$) are isomorphic. Some pre-ordered sets and lattices, every Boolean ring, and some other algebraic systems are determined up to isomorphism by their endomorphism semi-groups. The same is true for some modules and transformation semi-groups. Similar information about $X$ is carried by certain proper sub-semi-groups of $\operatorname{End}X$ (for example, the semi-groups of homeomorphic transformations of a topological space).
  
Some classes of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e03562024.png" /> (for example, topological spaces) can be characterized in this manner by their semi-groups of partial endomorphisms, that is, partial transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035620/e03562025.png" /> that are morphisms of their subobjects.
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Some classes of objects $X$ (for example, topological spaces) can be characterized in this manner by their semi-groups of partial endomorphisms, that is, partial transformations of $X$ that are morphisms of their subobjects.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.M. Gluskin,  "Transformation semigroups" , ''Proc. 4-th All-Union Math. Congress'' , '''2''' , Leningrad  (1964)  pp. 3–9  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Zykov,  "The theory of finite graphs" , '''1''' , Novosibirsk  (1969)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K.D. Magill,  "A survey of semigroups of continuous selfmaps"  ''Semigroup Forum'' , '''11'''  (1975–1976)  pp. 189–282</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M. Petrich,  "Rings and semigroups" , Springer  (1974)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.M. Gluskin,  "Transformation semigroups" , ''Proc. 4-th All-Union Math. Congress'' , '''2''' , Leningrad  (1964)  pp. 3–9  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Zykov,  "The theory of finite graphs" , '''1''' , Novosibirsk  (1969)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K.D. Magill,  "A survey of semigroups of continuous selfmaps"  ''Semigroup Forum'' , '''11'''  (1975–1976)  pp. 189–282</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M. Petrich,  "Rings and semigroups" , Springer  (1974)</TD></TR></table>

Revision as of 10:01, 15 April 2014

A semi-group consisting of the endomorphisms of a certain object (a set $X$ endowed with some structure $\sigma$) with the operation of multiplication (performing transformations in succession). The object $X$ can be a vector space, a topological space, an algebraic system, a graph, etc.; it is usually regarded as an object of a certain category, and as a rule, the morphisms (cf. Morphism) in this category are the mappings preserving the relations of (linear or continuous transformations, homomorphisms, etc.). The set $\operatorname{End}X$ of all endomorphism of $X$ (that is, of morphisms to its subobjects) is a sub-semi-group of the semi-group $T_X$ of all transformations of $X$ (see Transformation semi-group).

The semi-group $\operatorname{End}X$ may include a considerable amount of information on the structure $\sigma$. For example, if $X$ and $Y$ are vector spaces of dimensions $\geq2$ over skew-fields $F$ and $H$, respectively, then if the semi-groups $\operatorname{End}X$ and $\operatorname{End}Y$ of their endomorphisms (that is, linear transformations) are isomorphic, it follows that $X$ and $Y$ (and in particular, $F$ and $H$) are isomorphic. Some pre-ordered sets and lattices, every Boolean ring, and some other algebraic systems are determined up to isomorphism by their endomorphism semi-groups. The same is true for some modules and transformation semi-groups. Similar information about $X$ is carried by certain proper sub-semi-groups of $\operatorname{End}X$ (for example, the semi-groups of homeomorphic transformations of a topological space).

Some classes of objects $X$ (for example, topological spaces) can be characterized in this manner by their semi-groups of partial endomorphisms, that is, partial transformations of $X$ that are morphisms of their subobjects.

References

[1] L.M. Gluskin, "Transformation semigroups" , Proc. 4-th All-Union Math. Congress , 2 , Leningrad (1964) pp. 3–9 (In Russian)
[2] A.A. Zykov, "The theory of finite graphs" , 1 , Novosibirsk (1969) (In Russian)
[3] K.D. Magill, "A survey of semigroups of continuous selfmaps" Semigroup Forum , 11 (1975–1976) pp. 189–282
[4] M. Petrich, "Rings and semigroups" , Springer (1974)
How to Cite This Entry:
Endomorphism semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Endomorphism_semi-group&oldid=31709
This article was adapted from an original article by L.M. Gluskin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article