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Theorems on transferring properties of local parts of a [[Model (in logic)|model (in logic)]] to the whole model, established by A.I. Mal'tsev. A system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062180/m0621801.png" /> of subsets of a set is called a local covering of it if each element of the set is contained in some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062180/m0621802.png" /> and any two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062180/m0621803.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062180/m0621804.png" /> are contained in a third subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062180/m0621805.png" />. Examples of local coverings are: the system of all finite subsets of a set, and the system of all finitely-generated subgroups of a given group. A model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062180/m0621806.png" /> locally has a property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062180/m0621807.png" /> if there is a local covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062180/m0621808.png" /> consisting of submodels with the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062180/m0621809.png" />. A local theorem holds for a property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062180/m06218010.png" /> (and a corresponding class of models) if every model locally having property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062180/m06218011.png" /> has the property in the large.
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Theorems on transferring properties of local parts of a [[Model (in logic)|model (in logic)]] to the whole model, established by A.I. Mal'tsev. A system $\{ M_i : I \in I \}$ of subsets of a set is called a local covering of it if each element of the set is contained in some $M_i$ and any two $M_i$, $M_j$ are contained in a third subset $M_k$. Examples of local coverings are: the system of all finite subsets of a set, and the system of all finitely-generated subgroups of a given group. A model $M$ locally has a property $\sigma$ if there is a local covering of $M$ consisting of submodels with the property $\sigma$. A local theorem holds for a property $\sigma$ (and a corresponding class of models) if every model locally having property $\sigma$ has the property in the large.
  
A source of a great variety of local theorems is the following fundamental local theorem of Mal'tsev (or the compactness theorem of the restricted predicate calculus) [[#References|[1]]]: If each finite subsystem of an infinite system of axioms of the restricted predicate calculus is consistent, then the whole system is consistent. Mal'tsev [[#References|[2]]] gave a general method for obtaining concrete local theorems in group theory with the help of the fundamental local theorem, thus making a major contribution to [[Model theory|model theory]]. Later, by improving the method itself, he proved [[#References|[3]]] a local theorem for any property described by so-called quasi-universal axioms. The question of the validity of a local theorem for a property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062180/m06218012.png" />, which had previously been investigated separately for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062180/m06218013.png" />, has thus been reduced to a common and quite  "grammatical"  question: Is it possible to describe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062180/m06218014.png" /> by universal axioms?
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A source of a great variety of local theorems is the following fundamental local theorem of Mal'tsev (or the compactness theorem of the restricted predicate calculus) [[#References|[1]]]: If each finite subsystem of an infinite system of axioms of the restricted predicate calculus is consistent, then the whole system is consistent. Mal'tsev [[#References|[2]]] gave a general method for obtaining concrete local theorems in group theory with the help of the fundamental local theorem, thus making a major contribution to [[Model theory|model theory]]. Later, by improving the method itself, he proved [[#References|[3]]] a local theorem for any property described by so-called quasi-universal axioms. The question of the validity of a local theorem for a property $\sigma$, which had previously been investigated separately for each $\sigma$, has thus been reduced to a common and quite  "grammatical"  question: Is it possible to describe $\sigma$ by universal axioms?
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Mal'tsev,  "Investigation in the realm of mathematical logic"  ''Mat. Sb.'' , '''1''' :  3  (1936)  pp. 323–336  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Mal'tsev,  "A general method for obtainging local theorems in group theory"  ''Uchen. Zap. Ivanovsk. Gos. Ped. Inst.'' , '''1''' :  1  (1941)  pp. 3–9  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Mal'tsev,  "Model correspondences"  ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''23''' :  3  (1959)  pp. 313–336  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.I. Kargapolov,  J.I. [Yu.I. Merzlyakov] Merzljakov,  "Fundamentals of the theory of groups" , Springer  (1979)  (Translated from Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Mal'tsev,  "Investigation in the realm of mathematical logic"  ''Mat. Sb.'' , '''1''' :  3  (1936)  pp. 323–336  (In Russian)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Mal'tsev,  "A general method for obtaining local theorems in group theory"  ''Uchen. Zap. Ivanovsk. Gos. Ped. Inst.'' , '''1''' :  1  (1941)  pp. 3–9  (In Russian)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Mal'tsev,  "Model correspondences"  ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''23''' :  3  (1959)  pp. 313–336  (In Russian)</TD></TR>
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<TR><TD valign="top">[4]</TD> <TD valign="top">  M.I. Kargapolov,  J.I. [Yu.I. Merzlyakov] Merzljakov,  "Fundamentals of the theory of groups" , Springer  (1979)  (Translated from Russian)</TD></TR>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.I. [A.I. Mal'tsev] Mal'cev,  "The metamathematics of algebraic systems, Collected papers 1936–1967" , North-Holland  (1971)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.C. Chang,  H.J. Keisler,  "Model theory" , North-Holland  (1973)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.I. [A.I. Mal'tsev] Mal'cev,  "The metamathematics of algebraic systems, Collected papers 1936–1967" , North-Holland  (1971)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  C.C. Chang,  H.J. Keisler,  "Model theory" , North-Holland  (1973)</TD></TR>
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</table>
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Revision as of 22:22, 26 October 2014

Theorems on transferring properties of local parts of a model (in logic) to the whole model, established by A.I. Mal'tsev. A system $\{ M_i : I \in I \}$ of subsets of a set is called a local covering of it if each element of the set is contained in some $M_i$ and any two $M_i$, $M_j$ are contained in a third subset $M_k$. Examples of local coverings are: the system of all finite subsets of a set, and the system of all finitely-generated subgroups of a given group. A model $M$ locally has a property $\sigma$ if there is a local covering of $M$ consisting of submodels with the property $\sigma$. A local theorem holds for a property $\sigma$ (and a corresponding class of models) if every model locally having property $\sigma$ has the property in the large.

A source of a great variety of local theorems is the following fundamental local theorem of Mal'tsev (or the compactness theorem of the restricted predicate calculus) [1]: If each finite subsystem of an infinite system of axioms of the restricted predicate calculus is consistent, then the whole system is consistent. Mal'tsev [2] gave a general method for obtaining concrete local theorems in group theory with the help of the fundamental local theorem, thus making a major contribution to model theory. Later, by improving the method itself, he proved [3] a local theorem for any property described by so-called quasi-universal axioms. The question of the validity of a local theorem for a property $\sigma$, which had previously been investigated separately for each $\sigma$, has thus been reduced to a common and quite "grammatical" question: Is it possible to describe $\sigma$ by universal axioms?

References

[1] A.I. Mal'tsev, "Investigation in the realm of mathematical logic" Mat. Sb. , 1 : 3 (1936) pp. 323–336 (In Russian)
[2] A.I. Mal'tsev, "A general method for obtaining local theorems in group theory" Uchen. Zap. Ivanovsk. Gos. Ped. Inst. , 1 : 1 (1941) pp. 3–9 (In Russian)
[3] A.I. Mal'tsev, "Model correspondences" Izv. Akad. Nauk. SSSR Ser. Mat. , 23 : 3 (1959) pp. 313–336 (In Russian)
[4] M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)


Comments

English translations of references [1], [2] and [3] may be found as Chapts. 1, 2 and 11 in [a1].

References

[a1] A.I. [A.I. Mal'tsev] Mal'cev, "The metamathematics of algebraic systems, Collected papers 1936–1967" , North-Holland (1971) (Translated from Russian)
[a2] C.C. Chang, H.J. Keisler, "Model theory" , North-Holland (1973)
How to Cite This Entry:
Mal'tsev local theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mal%27tsev_local_theorems&oldid=11296
This article was adapted from an original article by Yu.I. Merzlyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article