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''finitely-approximable semi-group''
 
''finitely-approximable semi-group''
  
A [[Semi-group|semi-group]] for any two distinct elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r0815301.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r0815302.png" /> of which there is a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r0815303.png" /> of it into a finite semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r0815304.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r0815305.png" />. The property of a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r0815306.png" /> being residually finite is equivalent to that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r0815307.png" /> being a subdirect product of finite semi-groups. Residual finiteness is one of the more important finiteness conditions (see [[Semi-group with a finiteness condition|Semi-group with a finiteness condition]]); it is closely connected with algorithmic problems (cf. [[Algorithmic problem|Algorithmic problem]]): if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r0815308.png" /> is a finitely-presented residually-finite semi-group, then there is an algorithm for solving the word problem in it. The residually-finite semi-groups include the free semi-groups, the free commutative semi-groups, the free nilpotent semi-groups of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r0815309.png" />, the free inverse semi-groups (as algebras with two operations), the semi-lattices, the finitely-generated commutative semi-groups [[#References|[1]]], the finitely-generated semi-groups of matrices over a nilpotent or commutative ring, and the finitely-generated regular semi-groups that are nilpotent of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153010.png" /> in the sense of Mal'tsev (see [[Nilpotent semi-group|Nilpotent semi-group]]) [[#References|[4]]]; see also [[Residually-finite group|Residually-finite group]].
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A [[Semi-group|semi-group]] for any two distinct elements $a$ and $b$ of which there is a homomorphism $\phi$ of it into a finite semi-group $S$ such that $\phi(a)\neq\phi(b)$. The property of a semi-group $S$ being residually finite is equivalent to that of $S$ being a subdirect product of finite semi-groups. Residual finiteness is one of the more important finiteness conditions (see [[Semi-group with a finiteness condition|Semi-group with a finiteness condition]]); it is closely connected with algorithmic problems (cf. [[Algorithmic problem|Algorithmic problem]]): if $S$ is a finitely-presented residually-finite semi-group, then there is an algorithm for solving the word problem in it. The residually-finite semi-groups include the free semi-groups, the free commutative semi-groups, the free nilpotent semi-groups of class $n$, the free inverse semi-groups (as algebras with two operations), the semi-lattices, the finitely-generated commutative semi-groups [[#References|[1]]], the finitely-generated semi-groups of matrices over a nilpotent or commutative ring, and the finitely-generated regular semi-groups that are nilpotent of class $n$ in the sense of Mal'tsev (see [[Nilpotent semi-group|Nilpotent semi-group]]) [[#References|[4]]]; see also [[Residually-finite group|Residually-finite group]].
  
The [[Direct product|direct product]], the [[Free product|free product]], the ordinal sum (see [[Band of semi-groups|Band of semi-groups]]), and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153011.png" />-direct union of an arbitrary set of residually-finite semi-groups are also residually-finite semi-groups. Other constructions do not, generally speaking, preserve residual finiteness. An ideal extension of a residually-finite semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153012.png" /> by an arbitrary residually-finite semi-group is residually-finite if, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153013.png" /> is reductive, that is, if any two distinct elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153014.png" /> induce distinct left and distinct right inner translations; in particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153015.png" /> is a cancellation or inverse semi-group. The semi-lattice of a family of reductive residually-finite semi-groups is a residually-finite semi-group.
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The [[Direct product|direct product]], the [[Free product|free product]], the ordinal sum (see [[Band of semi-groups|Band of semi-groups]]), and the $0$-direct union of an arbitrary set of residually-finite semi-groups are also residually-finite semi-groups. Other constructions do not, generally speaking, preserve residual finiteness. An ideal extension of a residually-finite semi-group $S$ by an arbitrary residually-finite semi-group is residually-finite if, for example, $S$ is reductive, that is, if any two distinct elements of $S$ induce distinct left and distinct right inner translations; in particular, if $S$ is a cancellation or inverse semi-group. The semi-lattice of a family of reductive residually-finite semi-groups is a residually-finite semi-group.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153016.png" /> is a residually-finite semi-group, then all maximal subgroups of it are residually finite. For certain types of semi-groups this necessary condition is also sufficient; such as: regular semi-groups with a finite number of idempotents in every principal factor [[#References|[2]]], Clifford inverse semi-groups, and completely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153017.png" />-simple semi-groups with a finite number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153018.png" />- or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153019.png" />-classes (see [[Green equivalence relations|Green equivalence relations]]). For a number of classes of semi-groups a characterization of the residually-finite semi-groups in them has been obtained in terms not using reduction to maximal subgroups.
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If $S$ is a residually-finite semi-group, then all maximal subgroups of it are residually finite. For certain types of semi-groups this necessary condition is also sufficient; such as: regular semi-groups with a finite number of idempotents in every principal factor [[#References|[2]]], Clifford inverse semi-groups, and completely $0$-simple semi-groups with a finite number of $\mathcal L$- or $\mathcal R$-classes (see [[Green equivalence relations|Green equivalence relations]]). For a number of classes of semi-groups a characterization of the residually-finite semi-groups in them has been obtained in terms not using reduction to maximal subgroups.
  
Varieties of residually-finite semi-groups have been characterized in several ways [[#References|[3]]]. One such is the following. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153022.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153023.png" /> be, respectively, the two-element left zero and right zero semi-groups, the semi-group with zero multiplication, and a semi-lattice, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153024.png" /> be the three-element semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153027.png" /> and the remaining products are equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153028.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153029.png" /> be the semi-group anti-isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153030.png" />. A variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153031.png" /> consists of residually-finite semi-groups if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153032.png" /> is generated by a subset of one of the following three sets: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153036.png" /> is a finite group with Abelian Sylow subgroups and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081530/r08153037.png" /> is a finite cyclic group.
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Varieties of residually-finite semi-groups have been characterized in several ways [[#References|[3]]]. One such is the following. Let $L$, $R$, $N$, and $I$ be, respectively, the two-element left zero and right zero semi-groups, the semi-group with zero multiplication, and a semi-lattice, let $P$ be the three-element semi-group $\{e,p,0\}$, where $e^2=e$, $ep=p$ and the remaining products are equal to $0$, and let $P^*$ be the semi-group anti-isomorphic to $P$. A variety $M$ consists of residually-finite semi-groups if and only if $M$ is generated by a subset of one of the following three sets: $\{L,R,N,I,G\}$, $\{R,P,C\}$, $\{L,P^*,C\}$, where $G$ is a finite group with Abelian Sylow subgroups and $C$ is a finite cyclic group.
  
 
====References====
 
====References====

Latest revision as of 16:42, 19 September 2014

finitely-approximable semi-group

A semi-group for any two distinct elements $a$ and $b$ of which there is a homomorphism $\phi$ of it into a finite semi-group $S$ such that $\phi(a)\neq\phi(b)$. The property of a semi-group $S$ being residually finite is equivalent to that of $S$ being a subdirect product of finite semi-groups. Residual finiteness is one of the more important finiteness conditions (see Semi-group with a finiteness condition); it is closely connected with algorithmic problems (cf. Algorithmic problem): if $S$ is a finitely-presented residually-finite semi-group, then there is an algorithm for solving the word problem in it. The residually-finite semi-groups include the free semi-groups, the free commutative semi-groups, the free nilpotent semi-groups of class $n$, the free inverse semi-groups (as algebras with two operations), the semi-lattices, the finitely-generated commutative semi-groups [1], the finitely-generated semi-groups of matrices over a nilpotent or commutative ring, and the finitely-generated regular semi-groups that are nilpotent of class $n$ in the sense of Mal'tsev (see Nilpotent semi-group) [4]; see also Residually-finite group.

The direct product, the free product, the ordinal sum (see Band of semi-groups), and the $0$-direct union of an arbitrary set of residually-finite semi-groups are also residually-finite semi-groups. Other constructions do not, generally speaking, preserve residual finiteness. An ideal extension of a residually-finite semi-group $S$ by an arbitrary residually-finite semi-group is residually-finite if, for example, $S$ is reductive, that is, if any two distinct elements of $S$ induce distinct left and distinct right inner translations; in particular, if $S$ is a cancellation or inverse semi-group. The semi-lattice of a family of reductive residually-finite semi-groups is a residually-finite semi-group.

If $S$ is a residually-finite semi-group, then all maximal subgroups of it are residually finite. For certain types of semi-groups this necessary condition is also sufficient; such as: regular semi-groups with a finite number of idempotents in every principal factor [2], Clifford inverse semi-groups, and completely $0$-simple semi-groups with a finite number of $\mathcal L$- or $\mathcal R$-classes (see Green equivalence relations). For a number of classes of semi-groups a characterization of the residually-finite semi-groups in them has been obtained in terms not using reduction to maximal subgroups.

Varieties of residually-finite semi-groups have been characterized in several ways [3]. One such is the following. Let $L$, $R$, $N$, and $I$ be, respectively, the two-element left zero and right zero semi-groups, the semi-group with zero multiplication, and a semi-lattice, let $P$ be the three-element semi-group $\{e,p,0\}$, where $e^2=e$, $ep=p$ and the remaining products are equal to $0$, and let $P^*$ be the semi-group anti-isomorphic to $P$. A variety $M$ consists of residually-finite semi-groups if and only if $M$ is generated by a subset of one of the following three sets: $\{L,R,N,I,G\}$, $\{R,P,C\}$, $\{L,P^*,C\}$, where $G$ is a finite group with Abelian Sylow subgroups and $C$ is a finite cyclic group.

References

[1] A.I. Mal'tsev, "Homomorphisms onto finite groups" Uchen. Zap. Ivanovsk. Ped. Inst. , 18 (1958) pp. 49–60 (In Russian)
[2] E.A. Golubov, "Finitely approximable regular semi-groups" Math. Notes , 17 : 3 (1975) pp. 247–251 Mat. Zam. , 17 : 3 (1975) pp. 423–432
[3] E.A. Golubov, M.V. Sapir, "Varieties of finitely approximable semigroups" Soviet Math. Dokl. , 20 : 4 (1979) pp. 828–832 Dokl. Akad. Nauk SSSR , 247 : 5 (1979) pp. 1037–1041
[4] G. Lallement, "On nilpotency and residual finiteness in semigroups" Pacific J. Math. , 42 : 3 (1972) pp. 693–700


Comments

References

[a1] P.M. Cohn, "Universal algebra" , Reidel (1981)
How to Cite This Entry:
Residually-finite semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Residually-finite_semi-group&oldid=14997
This article was adapted from an original article by E.A. GolubovL.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article