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A semi-group in which every finitely-generated sub-semi-group is finite. A locally finite semi-group is a [[Periodic semi-group|periodic semi-group]] (torsion semi-group). The converse is false: There are even torsion groups that are not locally finite (see [[Burnside problem|Burnside problem]]). Long before the solution of the Burnside problem for groups, examples had been constructed of semi-groups that are torsion but not locally finite in classes of semi-groups remote from groups; above all, in the class of nil semi-groups (cf. [[Nil semi-group|Nil semi-group]]). These are, for example, a free semi-group with two generators in the variety given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060420/l0604201.png" />, and a free semi-group with three generators in the variety given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060420/l0604202.png" />. Moreover, for a number of classes of semi-groups the conditions of periodicity and local finiteness are equivalent. A trivial example is given by commutative semi-groups. A band of locally finite semi-groups (see [[Band of semi-groups|Band of semi-groups]]) is itself a locally finite semi-group [[#References|[1]]]; moreover, a semi-group that has a decomposition into locally finite groups is a locally finite semi-group; in particular, a semi-group of idempotents (cf. [[Idempotents, semi-group of|Idempotents, semi-group of]]) is a locally finite semi-group [[#References|[7]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060420/l0604203.png" /> is such that any group satisfying the law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060420/l0604204.png" /> is locally finite, then any semi-group with the law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060420/l0604205.png" /> is locally finite [[#References|[6]]]. A semi-group that has a decomposition into locally finite semi-groups need not be a locally finite semi-group [[#References|[3]]], but if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060420/l0604206.png" /> is a congruence on a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060420/l0604207.png" /> such that the quotient semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060420/l0604208.png" /> is locally finite and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060420/l0604209.png" />-class that is a sub-semi-group is locally finite, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060420/l06042010.png" /> is a locally finite semi-group (see [[#References|[4]]], [[#References|[5]]]); in particular, an ideal extension of a locally finite semi-group by a locally finite semi-group is itself a locally finite semi-group. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060420/l06042011.png" /> is a periodic semi-group of matrices over a skew-field and all subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060420/l06042012.png" /> are locally finite, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060420/l06042013.png" /> is locally finite [[#References|[8]]], which implies that any periodic semi-group of matrices over an arbitrary field is locally finite.
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A semi-group in which every finitely-generated sub-semi-group is finite. A locally finite semi-group is a [[Periodic semi-group|periodic semi-group]] (torsion semi-group). The converse is false: There are even torsion groups that are not locally finite (see [[Burnside problem|Burnside problem]]). Long before the solution of the Burnside problem for groups, examples had been constructed of semi-groups that are torsion but not locally finite in classes of semi-groups remote from groups; above all, in the class of nil semi-groups (cf. [[Nil semi-group|Nil semi-group]]). These are, for example, a free semi-group with two generators in the variety given by $x^3=0$, and a free semi-group with three generators in the variety given by $x^2=0$. Moreover, for a number of classes of semi-groups the conditions of periodicity and local finiteness are equivalent. A trivial example is given by commutative semi-groups. A band of locally finite semi-groups (see [[Band of semi-groups|Band of semi-groups]]) is itself a locally finite semi-group [[#References|[1]]]; moreover, a semi-group that has a decomposition into locally finite groups is a locally finite semi-group; in particular, a semi-group of idempotents (cf. [[Idempotents, semi-group of|Idempotents, semi-group of]]) is a locally finite semi-group [[#References|[7]]]. If $n$ is such that any group satisfying the law $x^n=1$ is locally finite, then any semi-group with the law $x^{n+1}=x$ is locally finite [[#References|[6]]]. A semi-group that has a decomposition into locally finite semi-groups need not be a locally finite semi-group [[#References|[3]]], but if $\rho$ is a congruence on a semi-group $S$ such that the quotient semi-group $S/\rho$ is locally finite and every $\rho$-class that is a sub-semi-group is locally finite, then $S$ is a locally finite semi-group (see [[#References|[4]]], [[#References|[5]]]); in particular, an ideal extension of a locally finite semi-group by a locally finite semi-group is itself a locally finite semi-group. If $S$ is a periodic semi-group of matrices over a skew-field and all subgroups of $S$ are locally finite, then $S$ is locally finite [[#References|[8]]], which implies that any periodic semi-group of matrices over an arbitrary field is locally finite.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060420/l06042014.png" /> is a periodic inverse semi-group of matrices over a field and, moreover, the periods of all its elements (see [[Monogenic semi-group|Monogenic semi-group]]) are uniformly bounded and are not divided by the characteristic of the field, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060420/l06042015.png" /> is finite [[#References|[2]]].
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If $S$ is a periodic inverse semi-group of matrices over a field and, moreover, the periods of all its elements (see [[Monogenic semi-group|Monogenic semi-group]]) are uniformly bounded and are not divided by the characteristic of the field, then $S$ is finite [[#References|[2]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.N. Shevrin,  "On locally finite semigroups"  ''Soviet Math. Dokl.'' , '''6'''  (1965)  pp. 769–772  ''Dokl. Akad. Nauk SSSR'' , '''162'''  (1965)  pp. 770–773</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.B. Shneperman,  "Periodic inverse linear semigroups"  ''Vesci Akad. Nauk BSSR Ser. Fiz. Mat. Nauk.'' , '''4'''  (1976)  pp. 22–28  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  T.C. Brown,  "On locally finite semigroups"  ''Ukr. Math. J.'' , '''20'''  (1968)  pp. 631–636  ''Ukr. Mat. Zh.'' , '''20''' :  6  (1968)  pp. 732–738</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  T.C. Brown,  "A semigroup union of disjoint locally finite subsemigroups which is not locally finite"  ''Pacific J. Math.'' , '''22''' :  1  (1967)  pp. 11–14</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  T.C. Brown,  "An interesting combinatorial method in the theory of locally finite semigroups"  ''Pacific J. Math.'' , '''36''' :  2  (1971)  pp. 285–289</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J.A. Green,  D. Rees,  "On semi-groups in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060420/l06042016.png" />"  ''Proc. Cambridge Philos. Soc.'' , '''48''' :  1  (1952)  pp. 35–40</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  D. McLean,  "Idempotent semigroups"  ''Amer. Math. Monthly'' , '''61''' :  2  (1954)  pp. 110–113</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  R. McNaughton,  Y. Zalcstein,  "The Burnside problem for semigroups"  ''J. of Algebra'' , '''34''' :  2  (1975)  pp. 292–299</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.N. Shevrin,  "On locally finite semigroups"  ''Soviet Math. Dokl.'' , '''6'''  (1965)  pp. 769–772  ''Dokl. Akad. Nauk SSSR'' , '''162'''  (1965)  pp. 770–773</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.B. Shneperman,  "Periodic inverse linear semigroups"  ''Vesci Akad. Nauk BSSR Ser. Fiz. Mat. Nauk.'' , '''4'''  (1976)  pp. 22–28  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  T.C. Brown,  "On locally finite semigroups"  ''Ukr. Math. J.'' , '''20'''  (1968)  pp. 631–636  ''Ukr. Mat. Zh.'' , '''20''' :  6  (1968)  pp. 732–738</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  T.C. Brown,  "A semigroup union of disjoint locally finite subsemigroups which is not locally finite"  ''Pacific J. Math.'' , '''22''' :  1  (1967)  pp. 11–14</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  T.C. Brown,  "An interesting combinatorial method in the theory of locally finite semigroups"  ''Pacific J. Math.'' , '''36''' :  2  (1971)  pp. 285–289</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J.A. Green,  D. Rees,  "On semi-groups in which $x^r=x$"  ''Proc. Cambridge Philos. Soc.'' , '''48''' :  1  (1952)  pp. 35–40</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  D. McLean,  "Idempotent semigroups"  ''Amer. Math. Monthly'' , '''61''' :  2  (1954)  pp. 110–113</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  R. McNaughton,  Y. Zalcstein,  "The Burnside problem for semigroups"  ''J. of Algebra'' , '''34''' :  2  (1975)  pp. 292–299</TD></TR></table>

Latest revision as of 08:27, 3 October 2014

A semi-group in which every finitely-generated sub-semi-group is finite. A locally finite semi-group is a periodic semi-group (torsion semi-group). The converse is false: There are even torsion groups that are not locally finite (see Burnside problem). Long before the solution of the Burnside problem for groups, examples had been constructed of semi-groups that are torsion but not locally finite in classes of semi-groups remote from groups; above all, in the class of nil semi-groups (cf. Nil semi-group). These are, for example, a free semi-group with two generators in the variety given by $x^3=0$, and a free semi-group with three generators in the variety given by $x^2=0$. Moreover, for a number of classes of semi-groups the conditions of periodicity and local finiteness are equivalent. A trivial example is given by commutative semi-groups. A band of locally finite semi-groups (see Band of semi-groups) is itself a locally finite semi-group [1]; moreover, a semi-group that has a decomposition into locally finite groups is a locally finite semi-group; in particular, a semi-group of idempotents (cf. Idempotents, semi-group of) is a locally finite semi-group [7]. If $n$ is such that any group satisfying the law $x^n=1$ is locally finite, then any semi-group with the law $x^{n+1}=x$ is locally finite [6]. A semi-group that has a decomposition into locally finite semi-groups need not be a locally finite semi-group [3], but if $\rho$ is a congruence on a semi-group $S$ such that the quotient semi-group $S/\rho$ is locally finite and every $\rho$-class that is a sub-semi-group is locally finite, then $S$ is a locally finite semi-group (see [4], [5]); in particular, an ideal extension of a locally finite semi-group by a locally finite semi-group is itself a locally finite semi-group. If $S$ is a periodic semi-group of matrices over a skew-field and all subgroups of $S$ are locally finite, then $S$ is locally finite [8], which implies that any periodic semi-group of matrices over an arbitrary field is locally finite.

If $S$ is a periodic inverse semi-group of matrices over a field and, moreover, the periods of all its elements (see Monogenic semi-group) are uniformly bounded and are not divided by the characteristic of the field, then $S$ is finite [2].

References

[1] L.N. Shevrin, "On locally finite semigroups" Soviet Math. Dokl. , 6 (1965) pp. 769–772 Dokl. Akad. Nauk SSSR , 162 (1965) pp. 770–773
[2] L.B. Shneperman, "Periodic inverse linear semigroups" Vesci Akad. Nauk BSSR Ser. Fiz. Mat. Nauk. , 4 (1976) pp. 22–28 (In Russian)
[3] T.C. Brown, "On locally finite semigroups" Ukr. Math. J. , 20 (1968) pp. 631–636 Ukr. Mat. Zh. , 20 : 6 (1968) pp. 732–738
[4] T.C. Brown, "A semigroup union of disjoint locally finite subsemigroups which is not locally finite" Pacific J. Math. , 22 : 1 (1967) pp. 11–14
[5] T.C. Brown, "An interesting combinatorial method in the theory of locally finite semigroups" Pacific J. Math. , 36 : 2 (1971) pp. 285–289
[6] J.A. Green, D. Rees, "On semi-groups in which $x^r=x$" Proc. Cambridge Philos. Soc. , 48 : 1 (1952) pp. 35–40
[7] D. McLean, "Idempotent semigroups" Amer. Math. Monthly , 61 : 2 (1954) pp. 110–113
[8] R. McNaughton, Y. Zalcstein, "The Burnside problem for semigroups" J. of Algebra , 34 : 2 (1975) pp. 292–299
How to Cite This Entry:
Locally finite semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locally_finite_semi-group&oldid=15721
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article