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2014-11-06T15:28:13Z

78.91.19.39: /* References */ Fixed DOI of reference to Lawshe (1975)

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Essential supremum and essential infimum

2013-12-14T11:22:36Z

78.91.24.53: /* Properties */ This is wrong.

In [[mathematical logic]], '''second-order arithmetic''' is a collection of [[axiom]]atic systems that formalize the [[natural number]]s and their subsets. It is an alternative to [[axiomatic set theory]] as a [[foundation of mathematics|foundation]] for much, but not all, of mathematics. It was introduced by [[David Hilbert]] and [[Paul Bernays]] in their book [[Grundlagen der Mathematik]]. The standard axiomatization of second-order arithmetic is denoted Z2.

Second-order arithmetic includes, but is significantly stronger than, its [[first order logic|first-order]] counterpart [[Peano arithmetic]]. Unlike Peano arithmetic, second-order arithmetic allows [[quantification]] over sets of numbers as well as numbers themselves. Because [[real number]]s can be represented as ([[infinite set|infinite]]) sets of natural numbers in well-known ways, and because second order arithmetic allows [[quantification]] over such sets, it is possible to formalize the [[real number]]s in second-order arithmetic. For this reason, second-order arithmetic is sometimes called “[[mathematical analysis|analysis]]”.

Second-order arithmetic can also be seen as a weak version of [[set theory]] in which every element is either a natural number or a set of natural numbers. Although it is much weaker than [[Zermelo-Fraenkel set theory]], second-order arithmetic can prove essentially all of the results of [[classical mathematics]] expressible in its language.

A '''subsystem of second-order arithmetic''' is a theory in the language of second-order arithmetic each axiom of which is a theorem of full second-order arithmetic (Z2). Such subsystems are essential to [[reverse mathematics]], a research program investigating how much of classical mathematics can be derived in certain weak subsystems of varying strength. Much of core mathematics can be formalized in these weak subsystems, some of which are defined below. Reverse mathematics also clarifies the extent and manner in which classical mathematics is [[nonconstructive]].

==Definition==
===Syntax===
The language of second-order arithmetic is two-sorted. The first sort of [[Term (mathematics)|terms]] and [[Variable (mathematics)|variables]], usually denoted by lower case letters, consists of [[individual]]s, whose intended interpretation is as natural numbers. The other sort of variables, variously called “set variables,” “class variables,” or even “predicates” are usually denoted by upper case letters. They refer to classes/predicates/properties of individuals, and so can be thought of as sets of natural numbers. Both individuals and set variables can be quantified universally or existentially. A formula with no [[bound variable|bound]] set variables (that is, no quantifiers over set variables) is called '''arithmetical'''. An arithmetical formula may have free set variables and bound individual variables.

Individual terms are formed from the constant 0, the unary function ''S'' (the ''[[successor function]]''), and the binary operations + and · (addition and multiplication). The successor function adds 1 (=''S''0) to its input. The relations = (equality) and < (comparison of natural numbers) relate two individuals, whereas the relation ∈ (membership) relates an individual and a set (or class).

For example, <math>\forall n (n\in X \rightarrow Sn \in X)</math>, is a [[well-formed formula]] of second-order arithmetic that is arithmetical, has one free set variable ''X'' and one bound individual variable ''n'' (but no bound set variables, as is required of an arithmetical formula)—whereas <math>\exists X \forall n(n\in X \leftrightarrow n < SSSSSS0\cdot SSSSSSS0)</math> is a well-formed formula that is not arithmetical with one bound set variable ''X'' and one bound individual variable ''n''.

===Semantics===
Several different interpretations of the quantifiers are possible. If second-order arithmetic is studied using the full semantics of [[second-order logic]] then the set quantifiers range over all subsets of the range of the number variables. If second-order arithmetic is formalized using the semantics of [[first-order logic]] then any model includes a domain for the set variables to range over, and this domain may be a proper subset of the full powerset of the domain of number variables.

Although second-order arithmetic was originally studied using full second-order semantics, the vast majority of current research treats second-order arithmetic in [[first-order predicate calculus]]. This is because the model theory of subsystems of second-order arithmetic is more interesting in the setting of first-order logic.

===Axioms===
====Basic====
The following axioms are known as the ''basic axioms'', or sometimes the ''Robinson axioms.'' The resulting [[first-order theory]], known as [[Robinson arithmetic]], is essentially [[Peano arithmetic]] without induction. The [[domain of discourse]] for the [[quantification|quantified variable]]s is the [[natural number]]s, collectively denoted by '''N''', and including the distinguished member <math>\ 0</math>, called "[[zero]]."

The primitive functions are the unary [[successor function]], denoted by [[prefix]] <math>\ S,</math>, and two [[binary operation]]s, [[addition]] and [[multiplication]], denoted by [[infix]] "+" and "<math> \cdot</math>", respectively. There is also a primitive [[binary relation]] called [[order relation|order]], denoted by infix "<".

Axioms governing the [[successor function]] and [[zero]]:

:1. <math>\forall m [Sm=0 \rightarrow \bot].</math> (“the successor of a natural number is never zero”)

:2. <math>\forall m \forall n [Sm=Sn \rightarrow m=n].</math> (“the successor function is [[Injective function|injective]]”)

:3. <math>\forall n [0=n \lor \exists m [Sm=n] ].</math> (“every natural number is zero or a successor”)

[[Addition]] defined [[recursion|recursively]]:

:4. <math>\forall m [m+0=m].</math>

:5. <math>\forall m \forall n [m+Sn = S(m+n)].</math>

[[Multiplication]] defined recursively:

:6. <math>\forall m [m\cdot 0 = 0].</math>

:7. <math>\forall m \forall n [m \cdot Sn = (m\cdot n)+m].</math>

Axioms governing the [[order relation]] "<":

:8. <math>\forall m [m<0 \rightarrow \bot].</math> (“no natural number is smaller than zero”)

:9. <math>\forall m [m<Sn \leftrightarrow (m<n \lor m=n)].</math>

:10. <math>\forall n [0=n \lor 0<n].</math> (“every natural number is zero or bigger than zero”)

:11. <math>\forall m \forall n [(Sm<n \lor Sm=n) \leftrightarrow m<n].</math>

These axioms are all [[first order logic|first order statements]]. That is, all variables range over the [[natural number]]s and not [[set theory|sets]] thereof, a fact even stronger than their being arithmetical. Moreover, there is but one [[existential quantifier]], in axiom 3. Axioms 1 and 2, together with an [[Peano axioms|axiom schema of induction]] make up the usual [[Peano axioms|Peano-Dedekind]] definition of '''N'''. Adding to these axioms any sort of [[Peano axioms|axiom schema of induction]] makes redundant the axioms 3, 10, and 11.

====Induction and comprehension schema====
If φ(''n'') is a formula of second-order arithmetic with a free number variable ''n'' and possible other free number or set variables (written ''m''• and ''X''•), the '''induction axiom''' for φ is the axiom:

:<math>\forall m_\bullet \forall X_\bullet ((\varphi(0) \land \forall n (\varphi(n) \rightarrow \varphi(Sn)) \rightarrow \forall n \varphi(n))</math>

The ('''full''') '''second-order induction scheme''' consists of all instances of this axiom, over all second-order formulas.

One particularly important instance of the induction scheme is when φ is the formula “<math>n \in X</math>” expressing the fact that ''n'' is a member of ''X'' (''X'' being a free set variable): in this case, the induction axiom for φ is

:<math>\forall X ((0\in X \land \forall n (n\in X \rightarrow Sn\in X)) \rightarrow \forall n (n\in X))</math>

This sentence is called the '''second-order induction axiom'''.

Returning to the case where φ(''n'') is a formula with a free variable ''n'' and possibly other free variables, we define the '''comprehension axiom''' for φ to be:

:<math>\forall m_\bullet \forall X_\bullet \exists Z \forall n (n\in Z \leftrightarrow \varphi(n))</math>

Essentially, this allows us to form the set <math>Z = \{ n | \varphi(n) \}</math> of natural numbers satisfying φ(''n''). There is a technical restriction that the formula φ may not contain the variable ''Z'', for otherwise the formula <math>n \not \in Z</math> would lead to the comprehension axiom
:<math>\exists Z \forall n ( n \in Z \leftrightarrow n \not \in Z)</math>,
which is inconsistent. This convention is assumed in the remainder of this article.

===The full system===
The formal theory of '''second-order arithmetic''' (in the language of second-order arithmetic) consists of the basic axioms, the comprehension axiom for every formula φ, (arithmetic or otherwise) and the second-order induction axiom. This theory is sometimes called ''full second order arithmetic'' to distinguish it from its subsystems, defined below. Second-order semantics eliminates the need for the comprehension axiom, because these semantics imply that every possible set exists.

In the presence of the unrestricted comprehension scheme, the single second-order induction axiom implies each instance of the full induction scheme. Subsystems that limit comprehension in some way may offset this limitation by including part of the induction scheme. Examples of such systems are provided below.

==Models of second-order arithmetic==
Recall that we view second-order arithmetic as a theory in first-order predicate calculus. Thus a '''model''' <math>\mathcal{M}</math> of the language of second-order arithmetic consists of a set ''M'' (which forms the range of individual variables) together with a constant 0 (an element of ''M''),
a function ''S'' from ''M'' to ''M'', two binary operations + and · on ''M'', a binary relation < on ''M'', and a collection ''D'' of subsets of ''M'', which is the range of the set variables. By omitting ''D'' we obtain a model of the language of first order arithmetic.

When ''D'' is the full powerset of ''M'', the model <math>\mathcal{M}</math> is called a '''full model'''. The use of full second-order semantics is equivalent to limiting the models of second-order arithmetic to the full models. In fact, the axioms of second-order arithmetic have only one full model. This follows from the fact that the axioms of [[Peano arithmetic]] with the second-order induction axiom have only one model under second-order semantics.

When ''M'' is the usual set of natural numbers with its usual operations, <math>\mathcal{M}</math> is called an '''ω-model'''. In this case we may identify the model with ''D'', its collection of sets of naturals, because this set is enough to completely determine an ω-model.

The unique full <math>\omega</math>-model, which is the usual set of natural numbers with its usual structure and all its subsets, is called the '''intended''' or '''standard''' model of second-order arithmetic.

==Definable functions of second-order arithmetic==
The first-order functions that are provably total in second-order arithmetic are precisely the same as those representable in [[system F]] (Girard ''et al.'', 1987, pp. 122–123). Almost equivalently, system F is the theory of functionals corresponding to second-order arithmetic in a manner parallel to how Gödel's [[system T]] corresponds to first-order arithmetic in the [[Dialectica interpretation]].

==Subsystems of second-order arithmetic==
{{main|reverse mathematics}}

There are many named subsystems of second-order arithmetic.

A subscript 0 in the name of a subsystem indicates that it includes only
a restricted portion of the full second-order induction scheme (Friedman 1976). Such a restriction lowers the [[proof-theoretic strength]] of the system significantly. For example, the system ACA0 described below is [[equiconsistency|equiconsistent]] with [[Peano arithmetic]]. The corresponding theory ACA, consisting of ACA0 plus the full second-order induction scheme, is stronger than Peano arithmetic.

===Arithmetical comprehension===
Many of the well-studied subsystems are related to closure properties of models. For example, it can be shown that every ω-model of full second-order arithmetic is closed under [[Turing jump]], but not every ω-model closed under Turing jump is a model of full second-order arithmetic. We may ask whether there is a subsystem of second-order arithmetic satisfied by every ω-model that is closed under Turing jump and satisfies some other, more mild, closure conditions.
The subsystem just described is called <math>\mathrm{ACA}_0</math>.

<math>\mathrm{ACA}_0</math> is defined as the theory consisting of the basic axioms, the '''arithmetical comprehension axiom''' scheme, in other words the comprehension axiom for every ''arithmetical'' formula φ, and the ordinary second-order induction axiom; again, we could also choose to include the arithmetical induction axiom scheme, in other words the induction axiom for every arithmetical formula φ, without making a difference.

It can be seen that a collection S of subsets of ω determines an ω-model of <math>\mathrm{ACA}_0</math> if and only if S is closed under Turing jump, Turing reducibility, and Turing join.

The subscript 0 in <math>\mathrm{ACA}_0</math> indicates that we have not included every instance of the induction axiom in this subsystem. This makes no difference when we study only ω-models, which automatically satisfy every instance of the induction axiom. It is of crucial importance, however, when we study models that are not ω-models. The system consisting of <math>\mathrm{ACA}_0</math> plus induction for all formulas is sometimes called <math>\mathrm{ACA}</math>.

The system <math>\mathrm{ACA}_0</math> is a conservative extension of '''first-order arithmetic''' (or first-order Peano axioms), defined as the basic axioms, plus the first order induction axiom scheme (for all formulas φ involving no class variables at all, bound or otherwise), in the language of first order arithmetic (which does not permit class variables at all). In particular it has the same [[Ordinal analysis|proof-theoretic ordinal]] ε0 as first-order arithmetic, owing to the limited induction schema.

===The arithmetical hierarchy for formulas===
{{main|Arithmetical hierarchy}}

To define a second subsystem, we will need a bit more terminology.

A formula is called ''bounded arithmetical'', or Δ00, when all its quantifiers are of the form ∀''n''<''t'' or ∃''n''<''t'' (where ''n'' is the individual variable being quantified and ''t'' is an individual term), where

:<math>\forall n<t(\cdots)</math>

stands for

:<math>\forall n(n<t \rightarrow \cdots)</math>

and

:<math>\exists n<t(\cdots)</math>

stands for

:<math>\exists n(n<t \land \cdots)</math>.

A formula is called Σ01 (or sometimes Σ1), respectively Π01 (or sometimes Π1) when it of the form ∃''m''•(φ), respectively ∀''m''•(φ) where φ is a bounded arithmetical formula and ''m'' is an individual variable (that is free in φ). More generally, a formula is called Σ0''n'', respectively Π0''n'' when it is obtained by adding existential, respectively universal, individual quantifiers to a Π0''n''−1, respectively Σ0''n''−1 formula (and Σ00 and Π00 are all equivalent to Δ00). Note that by construction all these formulas are arithmetical (no class variables are ever bound) and, in fact, by putting the formula in [[Skolem prenex form]] one can see that every arithmetical formula is equivalent to a Σ0''n'' or Π0''n'' formula for all large enough ''n''.

===Recursive comprehension===
The subsystem <math>\mathrm{RCA}_0</math> is an even weaker system than <math>\mathrm{ACA}_0</math> and is often used as the base system in [[reverse mathematics]]. It consists of: the basic axioms, the Σ01 induction scheme, and the Δ01 comprehension scheme. The former term is clear: the Σ01 induction scheme is the induction axiom for every Σ01 formula φ. The term “Δ01 comprehension” requires a little more explaining, however: there is no such thing as a Δ01 formula (the ''intended'' meaning is a formula that is both Σ01 and Π01), but we are instead postulating the comprehension axiom for every Σ01 formula ''subject to the condition'' that it is equivalent to a Π01 formula, in other words, for every Σ01 formula φ and every Π01 formula ψ we postulate

:<math>\forall m \forall X ((\forall n (\varphi(n) \leftrightarrow \psi(n))) \rightarrow \exists Z \forall n (n\in Z \leftrightarrow \varphi(n)))</math>

The set of first-order consequences of <math>\mathrm{RCA}_0</math> is the same as those of the subsystem IΣ1 of Peano arithmetic in which induction is restricted to Σ01 formulas. In turn, IΣ1 is conservative over [[primitive recursive arithmetic]] (PRA) for <math>\Pi^0_2</math> sentences. Moreover, the proof-theoretic ordinal of <math>\mathrm{RCA}_0</math> is ωω, the same as that of PRA.

It can be seen that a collection S of subsets of ω determines an ω-model of <math>\mathrm{RCA}_0</math>
if and only if S is closed under Turing reducibility and Turing join. In particular, the collection of all computable subsets of ω gives an ω-model of <math>\mathrm{RCA}_0</math>. This is the motivation behind the name of this system—if a set can be proved to exist using <math>\mathrm{RCA}_0</math>, then the set is computable (i.e. recursive).

=== Weaker systems ===
Sometimes an even weaker system than <math>\mathrm{RCA}_0</math> is desired. One such system is defined as follows: one must first augment the language of arithmetic with an exponential function (in stronger systems the exponential can be defined in terms of addition and multiplication by the usual trick, but when the system becomes too weak this is no longer possible) and the basic axioms by the obvious axioms defining exponentiation inductively from multiplication; then the system consists of the (enriched) basic axioms, plus Δ01 comprehension plus Δ00 induction.

===Stronger systems===
Much as we have defined Σ''n'' and Π''n'' (or, more accurately, Σ0''n'' and Π0''n'') formulae, we can define Σ1''n'' and Π1''n'' formulae in the following way: a Δ10 (or Σ10 or Π10) formula is just an arithmetical formula, and a Σ1''n'', respectively Π1''n'', formula is obtained by adding existential, respectively universal, class quantifiers in front of a Π1''n''−1, respectively Σ1''n''−1.

It is not too hard to see that over a not too weak system, any formula of second-order arithmetic is equivalent to a Σ1''n'' or Π1''n'' formula for all large enough ''n''. The system '''Π11-comprehension''' is the system consisting of the basic axioms, plus the ordinary second-order induction axiom and the comprehension axiom for every Π11 formula φ. It is an easy exercise to show that this is actually equivalent to Σ11-comprehension (on the other hand, Δ11-comprehension, defined by the same trick as introduced earlier for Δ01 comprehension, is actually weaker).

== Projective Determinacy ==
[[Projective determinacy]] is the assertion that every two-player perfect information game with moves being integers, game length ω and projective payoff set is determined, that is one of the players has a winning strategy. (The first player wins the game if the play belongs to the payoff set; otherwise, the second player wins.) A set is projective iff (as a predicate) it is expressible by a formula in the language of second order arithmetic, allowing real numbers as parameters, so projective determinacy is expressible as a schema in the language of Z2.

Many natural propositions expressible in the language of second order arithmetic are independent of Z2 and even [[ZFC]] but are provable from projective determinacy. Examples include coanalytic [[Perfect_set_property|perfect subset property]], measurability and the [[property of Baire]] for <math>\Sigma^1_2</math> sets, <math>\Pi^1_3</math> [[Uniformization_(set_theory)|uniformization]], etc. Over a weak base theory (such as RCA0), projective determinacy implies comprehension and provides an essentially complete theory of second order arithmetic — natural statements in the language of Z2 that are independent of Z2 with projective determinacy are hard to find. <ref>{{cite journal | author = W. Hugh Woodin | title = The Continuum Hypothesis, Part I | journal = Notices of the American Mathematical Society | volume = 48 | issue = 6 | year = 2001}}</ref>

ZFC + {there are ''n'' [[Woodin cardinal]]s: ''n'' is a natural number} is conservative over Z2 with projective determinacy, that is a statement in the language of second order arithmetic is provable in Z2 with projective determinacy iff its translation into the language of set theory is provable in ZFC + {there are ''n'' Woodin cardinals: ''n''∈N}.

==Coding mathematics in second-order arithmetic==
Second-order arithmetic allows us to speak directly (without coding) of natural numbers and sets of natural numbers. Pairs of natural numbers can be coded in the usual way as natural numbers, so arbitrary [[integer]]s or [[rational number]]s are first-class citizens in the same manner as natural numbers. [[Function (mathematics)|Functions]] between these sets can be encoded as sets of pairs, and hence as [[subset]]s of the natural numbers, without difficulty. [[Real number]]s can be defined as [[Cauchy sequence]]s of [[rational number]]s, but for technical reasons not discussed here, it is preferable (in the weak axiom systems above) to constrain the convergence rate (say by requiring that the distance between the ''n''-th and (''n''+1)-th term be less than 2−''n''). These systems cannot speak of real functions, or subsets of the reals. Nevertheless, [[continuous function|continuous]] real functions are legitimate objects of study, since they are defined by their values on the rationals. Moreover, a related trick makes it possible to speak of [[open subset]]s of the reals. Even [[Borel set]]s of reals can be coded in the language of second-order arithmetic, although doing so is a bit tricky.

==References==
*Burgess, John P., 2005. ''Fixing Frege''. Princeton University Press.
*Buss, S. R., ''Handbook of proof theory'' ISBN 0-444-89840-9
*Friedman, Harvey. "Systems of second order arithmetic with restricted induction," I, II (Abstracts). ''Journal of Symbolic Logic'', v.41, pp. 557-- 559, 1976. [http://www.jstor.org/stable/2272259 JStor]
*Girard, Lafont and Taylor, 1987. [http://www.monad.me.uk/stable/Proofs%2BTypes.html Proofs and Types]. Cambridge University Press.
*{{Citation | last1=Hilbert | first1=David | author1-link=David Hilbert | last2=Bernays | first2=Paul | title=Grundlagen der Mathematik | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Die Grundlehren der mathematischen Wissenschaften, Band 40, 50 | id={{MathSciNet | id = 0237246}} | year=1934}}
*{{Citation | last1=Simpson | first1=Stephen G. | title=Subsystems of second order arithmetic | url=http://www.math.psu.edu/simpson/sosoa/ | publisher=[[Cambridge University Press]] | edition=2nd | series=Perspectives in Logic | isbn=978-0-521-88439-6 | id={{MathSciNet | id = 2517689}} | year=2009}}
*[[Gaisi Takeuti]] (1975) ''Proof theory'' ISBN 0-444-10492-5
{{Reflist}}

==See also==
*[[Paris-Harrington theorem]]
*[[Reverse mathematics]]
*[[Presburger arithmetic]]
*[[Peano arithmetic]]
*[[Robinson arithmetic]]
*[[Second order logic]]

[[Category:Formal theories of arithmetic]]

Variational integrator

2012-09-03T09:34:01Z

78.91.45.155: In References: It's G. Wanner, not Warner

In the mathematical subject of [[group theory]], the '''Hanna Neumann conjecture''' is a statement about the [[rank of a group|rank]] of the intersection of two [[finitely generated group|finitely generated]] [[subgroup]]s of a [[free group]]. The conjecture was posed by [[Hanna Neumann]] in 1957.<ref name="HN57">Hanna Neumann. ''On the intersection of finitely generated free groups. Addendum.'' [[Publicationes Mathematicae Debrecen]], vol. 5 (1957), p. 128</ref> In 2011, the conjecture was proved independently by Igor Mineyev<ref name="proof">Igor Minevev,
[http://annals.math.princeton.edu/2012/175-1/p11/ "Submultiplicativity and the Hanna Neumann Conjecture."]
Ann. of Math., 175 (2012), no. 1, 393-414</ref> and Joel Friedman. <ref name="proof2">Joel Friedman,
[http://www.math.ubc.ca/~jf/pubs/web_stuff/shnc_memoirs.pdf "Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture."]
to appear in Memoirs of the AMS</ref>

==History==

The subject of the conjecture was originally motivated by a 1954 theorem of Howson<ref>A. G. Howson. ''On the intersection of finitely generated free groups.'' [[Journal of the London Mathematical Society]], vol. 29 (1954), pp. 428–434</ref> who proved that the intersection of any two [[finitely generated group|finitely generated]] [[subgroup]]s of a [[free group]] is always finitely generated, that is, has finite [[rank of a group|rank]]. In this paper Howson proved that if ''H'' and ''K'' are [[subgroup]]s of a free group ''F''(''X'') of finite ranks ''n'' ≥ 1 and ''m'' ≥ 1 then the rank ''s'' of ''H'' ∩ ''K'' satisfies:
:''s'' − 1 ≤ 2''mn'' − ''m'' − ''n''.

In a 1956 paper<ref>Hanna Neumann. ''On the intersection of finitely generated free groups.'' Publicationes Mathematicae Debrecen, vol. 4 (1956), 186–189.</ref> [[Hanna Neumann]] improved this bound by showing that :

:''s'' − 1 ≤ 2''mn'' − ''2m'' − ''n''.

In a 1957 addendum,<ref name="HN57"/> Hanna Neumann further improved this bound to show that under the above assumptions

:''s'' − 1 ≤ 2(''m'' − 1)(''n'' − 1).

She also conjectured that the factor of 2 in the above inequality is not necessary and that one always has

:''s'' − 1 ≤ (''m'' − 1)(''n'' − 1).

This statement became known as the ''Hanna Neumann conjecture''.

==Formal statement==

Let ''H'', ''K'' ≤ ''F''(''X'') be two nontrivial finitely generated subgroups of a [[free group]] ''F''(''X'') and let ''L'' = ''H'' ∩ ''K'' be the intersection of ''H'' and ''K''. The conjecture says that in this case

:rank(''L'') − 1 ≤ (rank(''H'') − 1)(rank(''K'') − 1).

Here for a group ''G'' the quantity rank(''G'') is the [[rank of a group|rank]] of ''G'', that is, the smallest size of a [[generating set of a group|generating set]] for ''G''.
Every [[subgroup]] of a [[free group]] is known to be [[free group|free]] itself and the [[rank of a group|rank]] of a [[free group]] is equal to the size of any free basis of that free group.

==Strengthened Hanna Neumann conjecture==

If ''H'', ''K'' ≤ ''G'' are two subgroups of a [[group (mathematics)|group]] ''G'' and if ''a'', ''b'' ∈ ''G'' define the same [[double coset]] ''HaK = HbK'' then the [[subgroup]]s ''H'' ∩ ''aKa''−1 and ''H'' ∩ ''bKb''−1 are [[Conjugacy class|conjugate]] in ''G'' and thus have the same [[rank of a group|rank]]. It is known that if ''H'', ''K'' ≤ ''F''(''X'') are [[finitely generated group|finitely generated]] subgroups of a finitely generated [[free group]] ''F''(''X'') then there exist at most finitely many double coset classes ''HaK'' in ''F''(''X'') such that ''H'' ∩ ''aKa''−1 ≠ {1}. Suppose that at least one such double coset exists and let ''a''1,...,''a''''n'' be all the distinct representatives of such double cosets. The ''strengthened Hanna Neumann conjecture'', formulated by her son [[Walter Neumann]] (1990),<ref name="WN">Walter Neumann. ''On intersections of finitely generated subgroups of free groups.'' Groups–Canberra 1989, pp. 161–170. Lecture Notes in Mathematics, vol. 1456, Springer, Berlin, 1990; ISBN 3-540-53475-X</ref> states that in this situation

:<math>\sum_{i=1}^n [{\rm rank}(H\cap a_iKa_{i}^{-1})-1] \le ({\rm rank}(H)-1)({\rm rank}(K)-1).</math>

==Partial results and other generalizations==

*In 1971 Burns improved<ref>Robert G. Burns.
[http://www.springerlink.com/content/u75hh74lu1053790/ ''On the intersection of finitely generated subgroups of a free group.'']
[[Mathematische Zeitschrift]], vol. 119 (1971), pp. 121–130.</ref> Hanna Neumann's 1957 bound and proved that under the same assumptions as in Hanna Neumann's paper one has

:''s'' ≤ 2''mn'' − 3''m'' − 2''n'' + 4.

*In a 1990 paper,<ref name="WN"/> Walter Neumann formulated the strengthened Hanna Neumann conjecture (see statement above).
*[[Gábor Tardos|Tardos]] (1992)<ref>Gábor Tardos. [http://www.springerlink.com/content/n013g5rx543x4748/ ''On the intersection of subgroups of a free group.'']
[[Inventiones Mathematicae]], vol. 108 (1992), no. 1, pp. 29–36.</ref> established the Hanna Neumann Conjecture for the case where at least one of the subgroups ''H'' and ''K'' of ''F''(''X'') has rank two. As most other approaches to the Hanna Neumann conjecture, Tardos used the technique of [[Stallings subgroup graph]]s<ref>John R. Stallings. [http://www.springerlink.com/content/mn2h645qw2058530/ ''Topology of finite graphs.''] [[Inventiones Mathematicae]], vol. 71 (1983), no. 3, pp. 551–565</ref> for analyzing subgroups of free groups and their intersections.
*Warren Dicks (1994)<ref>Warren Dicks. [http://www.springerlink.com/content/r526373840056u7q/ ''Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture.''] [[Inventiones Mathematicae]], vol. 117 (1994), no. 3, pp. 373–389</ref> established the equivalence of the strengthened Hanna Neumann conjecture and a graph-theoretic statement that he called the ''amalgamated graph conjecture''.
*Arzhantseva (2000) proved<ref>G. N. Arzhantseva. [http://www.ams.org/proc/2000-128-11/S0002-9939-00-05508-8/home.html ''A property of subgroups of infinite index in a free group''] [[Proceedings of the American Mathematical Society|Proc. Amer. Math. Soc.]] 128 (2000), 3205–3210.</ref> that if ''H'' is a finitely generated subgroup of infinite index in ''F''(''X''), then, in a certain statistical meaning, for a generic finitely generated subgroup <math>K</math> in <math>F(X)</math>, we have ''H'' ∩ ''gKg''−1 = {1} for all ''g'' in ''F''. Thus, the strengthened Hanna Neumann conjecture holds for every ''H'' and a generic ''K''.
*In 2001 Dicks and Formanek used this equivalence to prove the strengthened Hanna Neumann Conjecture in the case when one of the subgroups ''H'' and ''K'' of ''F''(''X'') has rank at most three.<ref>Warren Dicks, and Edward Formanek. ''The rank three case of the Hanna Neumann conjecture.'' Journal of Group Theory, vol. 4 (2001), no. 2, pp. 113–151</ref>
*Khan (2002)<ref>Bilal Khan. ''Positively generated subgroups of free groups and the Hanna Neumann conjecture.'' Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), 155–170,
Contemporary Mathematics, vol. 296, [[American Mathematical Society]], Providence, RI, 2002; ISBN 0-8218-2822-3</ref> and, independently, Meakin and Weil (2002),<ref>J. Meakin, and P. Weil. [http://www.springerlink.com/content/m742547j1g534g40/ ''Subgroups of free groups: a contribution to the Hanna Neumann conjecture.'']
Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000).
[[Geometriae Dedicata]], vol. 94 (2002), pp. 33–43.</ref> showed that the conclusion of the strengthened Hanna Neumann conjecture holds if one of the subgroups ''H'', ''K'' of ''F''(''X'') is ''positively generated'', that is, generated by a finite set of words that involve only elements of ''X'' but not of ''X''−1 as letters.
*Ivanov<ref>S. V. Ivanov. ''Intersecting free subgroups in free products of groups.'' International Journal of Algebra and Computation, vol. 11 (2001), no. 3, pp. 281–290</ref><ref>S. V. Ivanov. ''On the Kurosh rank of the intersection of subgroups in free products of groups''. [[Advances in Mathematics]], vol. 218 (2008), no. 2, pp. 465–484</ref> and, subsequently, Dicks and Ivanov,<ref>Warren Dicks, and S. V. Ivanov. ''On the intersection of free subgroups in free products of groups.'' Mathematical Proceedings of the Cambridge Philosophical Society, vol. 144 (2008), no. 3, pp. 511–534</ref> obtained analogs and generalizations of Hanna Neumann's results for the intersection of [[subgroup]]s ''H'' and ''K'' of a [[free product]] of several groups.
*Wise (2005) showed<ref>[http://blms.oxfordjournals.org.proxy2.library.uiuc.edu/cgi/content/abstract/37/5/697 ''The Coherence of One-Relator Groups with Torsion and the Hanna Neumann Conjecture.''] [[Bulletin of the London Mathematical Society]], vol. 37 (2005), no. 5, pp. 697–705</ref> that the strengthened Hanna Neumann conjecture implies another long-standing group-theoretic conjecture which says that every one-relator group with torsion is ''coherent'' (that is, every [[finitely generated group|finitely generated]] subgroup in such a group is [[finitely presented group|finitely presented]]).

==See also==
*[[Rank of a group]]
*[[Geometric group theory]]

==References==
{{reflist}}

[[Category:Group theory]]
[[Category:Geometric group theory]]

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2011-10-09T17:43:07Z

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