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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005022.png" /></td> </tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005022.png" /></td> </tr></table>
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005023.png" /> is defined to be the set of all subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005025.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005026.png" /> for some closed unbounded subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005028.png" />. (See [[Suslin hypothesis|Suslin hypothesis]] for the definitions of closed unbounded and stationary.) The equivalence class depends only on the isomorphism type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005029.png" /> and not on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005030.png" />-filtration, because any two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005031.png" />-filtrations agree on a closed unbounded subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005032.png" />; the equivalence classes of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005033.png" /> form a [[Boolean algebra|Boolean algebra]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005034.png" />, under the partial order induced by inclusion. The least element of this Boolean algebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005035.png" />, is the class of all non-stationary subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005036.png" />. It can be proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005037.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005038.png" /> is free (see [[#References|[a1]]]). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005039.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005040.png" />), every one of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005041.png" /> members of the Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005042.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005043.png" />-invariant of some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005044.png" />-free group of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005045.png" /> (see [[#References|[a6]]]). Assuming Gödel's [[axiom of constructibility]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005046.png" /> (see [[Gödel constructive set|Gödel constructive set]]), the same holds for all regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005047.png" /> which are not too large (e.g., less than the first inaccessible cardinal, or even the first Mahlo cardinal); in fact, a complete characterization, for any regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005048.png" />, of the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005049.png" /> can be given, assuming <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005050.png" /> (see [[#References|[a6]]] and [[#References|[a5]]]).
+
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005023.png" /> is defined to be the set of all subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005025.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005026.png" /> for some closed unbounded subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005028.png" />. (See [[Suslin hypothesis|Suslin hypothesis]] for the definitions of closed unbounded and stationary.) The equivalence class depends only on the isomorphism type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005029.png" /> and not on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005030.png" />-filtration, because any two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005031.png" />-filtrations agree on a closed unbounded subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005032.png" />; the equivalence classes of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005033.png" /> form a [[Boolean algebra|Boolean algebra]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005034.png" />, under the partial order induced by inclusion. The least element of this Boolean algebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005035.png" />, is the class of all non-stationary subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005036.png" />. It can be proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005037.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005038.png" /> is free (see [[#References|[a1]]]). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005039.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005040.png" />), every one of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005041.png" /> members of the Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005042.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005043.png" />-invariant of some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005044.png" />-free group of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005045.png" /> (see [[#References|[a6]]]). Assuming Gödel's [[axiom of constructibility]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005046.png" /> (see [[Gödel constructive set|Gödel constructive set]]), the same holds for all regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005047.png" /> which are not too large (e.g., less than the first [[inaccessible cardinal]], or even the first [[Mahlo cardinal]]); in fact, a complete characterization, for any regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005048.png" />, of the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005049.png" /> can be given, assuming <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005050.png" /> (see [[#References|[a6]]] and [[#References|[a5]]]).
  
 
Another <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005051.png" />-invariant can be defined for use in connection with the [[Whitehead problem|Whitehead problem]] in Abelian group theory, and its generalizations. In this case, for any Abelian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005054.png" /> is defined to be the equivalence class of
 
Another <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005051.png" />-invariant can be defined for use in connection with the [[Whitehead problem|Whitehead problem]] in Abelian group theory, and its generalizations. In this case, for any Abelian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110050/g11005054.png" /> is defined to be the equivalence class of

Latest revision as of 20:32, 4 December 2017

-invariant

An invariant associated to an uncountable Abelian group and taking values in a Boolean algebra. Two groups with different invariants are non-isomorphic, but the converse fails in general: groups with the same invariant need not be isomorphic. The invariant is most commonly defined for almost-free groups (groups such that every subgroup of strictly smaller cardinality is a free Abelian group). By a theorem of S. Shelah (see [a7]), such a group is free if it is of singular cardinality, so the invariant is defined for groups of regular cardinality (see Cardinal number). If is an Abelian group of regular uncountable cardinality , is said to be -free if and only if every subgroup of of cardinality is free. In that case can be written as the union of a continuous chain (called a -filtration) of free subgroups of cardinality : , where the continuity condition means that for every limit ordinal , . The -invariant of , denoted by or just , is defined to be the equivalence class, , of

is defined to be the set of all subsets of for which for some closed unbounded subset of . (See Suslin hypothesis for the definitions of closed unbounded and stationary.) The equivalence class depends only on the isomorphism type of and not on the choice of -filtration, because any two -filtrations agree on a closed unbounded subset of ; the equivalence classes of subsets of form a Boolean algebra, , under the partial order induced by inclusion. The least element of this Boolean algebra, , is the class of all non-stationary subsets of . It can be proved that if and only if is free (see [a1]). For (), every one of the members of the Boolean algebra is the -invariant of some -free group of cardinality (see [a6]). Assuming Gödel's axiom of constructibility, (see Gödel constructive set), the same holds for all regular which are not too large (e.g., less than the first inaccessible cardinal, or even the first Mahlo cardinal); in fact, a complete characterization, for any regular , of the range of can be given, assuming (see [a6] and [a5]).

Another -invariant can be defined for use in connection with the Whitehead problem in Abelian group theory, and its generalizations. In this case, for any Abelian groups and , is defined to be the equivalence class of

when is -free of cardinality and is written as the union, , of a -filtration. Then implies ; the converse holds for of cardinality at most , assuming (see [a2]).

Useful references for additional information are [a3] and [a4].

References

[a1] P.C. Eklof, "Methods of logic in abelian group theory" , Abelian Group Theory , Lecture Notes in Mathematics , 616 , Springer (1977) pp. 251–269
[a2] P.C. Eklof, "Homological algebra and set theory" Trans. Amer. Math. Soc. , 227 (1977) pp. 207–225
[a3] P.C. Eklof, "Set-theoretic methods: the uses of gamma invariants" , Abelian Groups , Lecture Notes in Pure and Appl. Math. , 146 , M. Dekker (1993) pp. 143–153
[a4] P.C. Eklof, A.H. Mekler, "Almost free modules" , North-Holland (1990)
[a5] P.C. Eklof, A.H. Mekler, S. Shelah, "Almost disjoint abelian groups" Israel J. Math. , 49 (1984) pp. 34–54
[a6] A.H. Mekler, "How to construct almost free groups" Canad. J. Math. , 32 (1980) pp. 1206–1228
[a7] S. Shelah, "A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals" Israel J. Math. , 21 (1975) pp. 319–349
How to Cite This Entry:
Gamma-invariant in the theory of Abelian groups. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Gamma-invariant_in_the_theory_of_Abelian_groups&oldid=35000
This article was adapted from an original article by P.C. Eklof (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article