View source for Endomorphism 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 $\operatorname{End}X$ 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====
<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>

[[Category:General algebraic systems]]