Bulletin of the Section of Logic
Volume 39:3/4 (2010), pp. 153–160
Ewa Graczyńska
DEPENDENCE SPACES
Abstract
This is a continuation of my lecture presented on 77th Workshop on General
Algebra, 24th Conference for Young Algebraists in Potsdam (Germany) on 21st
March 2009. The Steinitz exchange lemma is a basic theorem in linear algebra
used, for example, to show that any two bases for a finite-dimensional vector
space have the same number of elements. The result is named after the German
mathematician Ernst Steinitz1 .
1.
Introduction
According to F. Gécseg, H. Jürgensen [4] the result which is usually referred to as the ”Exchange Lemma”, states that for transitive dependence,
every independent set can be extended to form a basis. Our aim is to
discuss some interplay between the discussed notion of [5]–[6] and [3]–[4].
We present another proof of the result of N.J.S. Hughes [5] on Steinitz’
exchange theorem for infinite bases in connection with the notions of transitive dependence, independence and dimension as introduced in [3] and
[9]. In the proof we assume Kuratowski-Zorn’s Lemma, as a requirement
pointed in [4]. In our opinion the proof is simpler as those of [5].
We use a modification of the the notation of [5], [6] and [4]:
a, b, c, ..., x, y, z, ... (with or without suffices) to denote the elements of S
and A, B, C, ..., X, Y, Z, ... for subsets of S, X, Y,... denote a family of
subsets of S, n is always a positive integer.
1 AMS Mathematical Subject Classification 2000:
Primary: 00A05, 08A99. Secondary: 11J72, 11J85
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Ewa Graczyńska
A ∪ B denotes the union of sets A and B, A + B denotes the disjoint
union of A and B, A − B denotes the difference of A and B, i.e. is the set
of those elements of A which are not in B.
2.
Dependent and independent sets
The following definition is due to N.J.S. Hughes, invented in 1962 in [5]:
Definition 1. A set S is called a dependence space if there is defined a
set ∆, whose members are finite subsets of S, each containing at least 2
elements, and if the Transitivity Axiom is satisfied.
Definition 2. A set A is called directly dependent if A ∈ ∆.
Definition 3. An element x is called dependent on A and is denoted by
x ∼ ΣA if either x ∈ A or if there exist distinct elements x0 , x1 , ..., xn such
that
(1) {x0 , x1 , ..., xn } ∈ ∆
where x0 = x and x1 , ..., xn ∈ A
and directly dependent on {x} or {x0 , x1 , ..., xn }, respectively.
Definition 4. A set A is called dependent if (1) is satisfied for some distinct
elements x0 , x1 , ..., xn ∈ A, and otherwise A is independent.
Definition 5. If a set A is independent and for any x ∈ S, x ∼ ΣA, i.e. x
is dependent on A, then A is called a basis of S.
Definition 6. (TRANSITIVITY AXIOM:)
If x ∼ ΣA and for all a ∈ A, a ∼ ΣB, then x ∼ ΣB.
A similar definition of a dependence D was introduced in [3] and [4].
First we recall the definition of [4], p. 425 and [3]:
Definition 7. Let S be a set with dependence D and a ∈ S, A ⊆ S. The
element x is said to depend on A if a ∈ A or there is an independent subset
A0 ⊆ A such that A0 ∪ {a} is dependent in S.
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155
Lemma 8. In a dependence space S the following are equivalent for an
element x ∈ S and a subset A of S:
(i) x ∼ ΣA in the sense of [5] and
(ii) x depends of A in the sense of [4] and [3].
Proof. Assume (i), i.e. that x ∼ ΣA in the sense of [5]. Therefore,
x ∈ A or there exists elements x0 = x and x1 , ..., xn ∈ A such that
{x0 , x1 , ..., xn } ∈ ∆. If the set {x1 , ..., xn } is dependent in S in the sense
of [5], then there exists its finite subset, say A0 ⊆ {x1 , ..., xn }, such that
A0 ∈ ∆, i.e. A0 is directly dependent in S. If we assume that (ii) is not
satisfied, i.e. for all independent subsets A” of A, the set {x} ∪ A” is independent in S, then A” is independent, for all A” ⊆ A, a contradiction for
A0 = A”. We conclude (ii).
Assume now (ii), i.e. that x depends of A in the sense of [4] and [3].
Thus x ∈ A or there exists an independent subset A0 ⊆ A such that A0 ∪{x}
is dependent in S. In the second case we get that {x} ∪ A0 is dependent in
S, i.e. there exists x0 = x and x1 , ..., xn ∈ A0 , such that x0 , x1 , ..., xn ∈ ∆,
as A0 is independent. We conclude (i). Following ideas of [3] and [4] we accept the following:
Definition 9. The span < X > of a subset X of S is the set of all elements
of S which depends on X, i.e. x ∈< X > iff x ∼ ΣX. A subset X of S
is called closed if X =< X >, and a dependence D is called transitive, if
< X >=<< X >>, for all subsets X of S.
First we note, that in a dependence space D, the notion of transitive
dependence of [3] and [4] is equivalent to those of [5].
Theorem 10. Given a dependence space S satisfying the Transitivity Axiom in the sense of [5]. Then dependence space S is transitive in the sense
of [3] and [4]. And vice versa, if a dependence space S is transitive in the
sense of [3] and [4], then the Transitivity Axiom of [5] is satisfied.
Proof. Let a transitive axiom of [5] be satisfied in S and let X ⊆ S.
X ⊆< X > and therefore < X >⊆<< X >>, by Remark 3.6 of [3]. Now
let x ∈<< X >>, i.e. x ∼ Σ < X >. But y ∼ ΣX, for all y ∈< X >.
Thus by the Transitivity Axiom x ∼ ΣX, i.e. x ∈< X >. We get that
< X >=<< X >>.
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Ewa Graczyńska
Now, assume that a dependence is transitive in the sense of [3], [4], i.e.
S be a transitive dependence space, i.e. << X >>=< X > for all X ⊆ S.
Let x ∼ ΣA and for all a ∈ A, a ∼ ΣB. Thus x ∈ A or the set {x} ∪ A is
dependent, i.e. x ∈< A >. Moreover, a ∈< B >, for all a ∈ A. Therefore
{a} ⊆< B >, i.e. <Sa >⊆<< B >>=<
S B >, by the Remark 3.6 of [4]. In
consequence: A = {a : a ∈ A} ⊆ {< a >: a ∈ A} ⊆< B > and thus:
< A >⊆<< B >>=< B >. We get that x ∈< B >, i.e. x ∼ ΣB, i.e. the
Transitivity Axiom is satisfied. Note, that the following well known properties (see [3], [9] or [4]) are
satisfied in a dependence space:
Proposition 11. (2) Any subset of an independent set A is independent.
(3) A basis is a maximal independent set of S and vice versa.
(4) A basis is a minimal subset of S which spans S and vice versa.
(5) The family (X, ⊆) of all independent subsets of S is partially ordered
by the set-theoretical inclusion. Shortly we say that X is an ordered set
(a po-set).
(6) Any superset of a dependent set of S is dependent.
The following are well known (see [5] and [4], [3]):
Example 12. Consider the two-dimensional vector space R2 . ∆ is defined
as follows. A subset X of R2 containing two parallel vectors is always
dependent. A finite subset X of R2 with more that 2 elements is always
dependent. The one element subset {0} is defined as dependent. Transitivity Axiom is satisfied in such dependance space S. Independent subsets
contain at most two non parallel vectors. Bases are bases of R2 in the
classical sense.
Example 13. Let K denotes the set of some (at least two) colours, S =
C(K) is the set of all sequences of K. ∆ = Cf in (K) be the set containing
all finite at lest two-element sequences of elements of K with at least one
repetition of colours. Then the Transitivity Axiom is satisfied for such
defined dependence space S.
Example 14. Let C(N) denotes the set of all sequences of natural numbers
and ∆ = Cf in (N) be the set containing all finite at lest two-element se-
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157
quences of elements of N with at least one repetition. Then the Transitivity
Axiom is satisfied for such defined dependence space S.
Example 15. A graph is an abstract representation of a set of objects
where some pairs of the objects are connected by links. The interconnected
objects are represented by mathematical abstractions called vertices, and
the links that connect some pairs of vertices are called edges. Typically, a
graph is depicted in diagrammatic form as a set of dots for the vertices,
joined by lines or curves for the edges. For a graph, call a finite subset A
of its vertices directly dependent if it has at least two elements which are
connected. Then for a vertex x, put x ∼ ΣA if there exists a link between
x and a vertex a in A. Then the Transitivity Axiom for such a dependence
space is satisfied.
3.
Po-set of independent sets
Following K. Kuratowski and A. Mostowski [8] p. 241, a po-set (X, ⊆) is
called closed if for every chain of sets A ⊆ P (X) there exists ∪A in X, i.e.
A has the supremum in (X, ⊆).
Theorem 16. The po-set (X, ⊆) of all independent subsets of S is closed.
Proof. Let A be a chain of independent subsets of S, i.e. A ⊆ P (X),
and for all A, B ∈ A (A ⊆ B) or (B ⊆ A). We show that the set ∪A is
independent. Otherwise there exist elements {x0 , x1 , ..., xn } ∈ ∆ such that
xi ∈ ∪A, for i = 0, ..., n. Therefore there exists a set A ∈ A such that
xi ∈ A for all i = 0, ..., n. We conclude that A is dependent, A ∈ A, a
contradiction. Theorem 17. The po-set (C(X), ⊆) of all independent subsets of S is an
algebraic closure system.
Proof. Let A be a directed family of independent subsets of S, i.e.
A ⊆ P (X), and for all A, B ∈ A there exists a C ∈ A such that (A ⊆ C)
and (B ⊆ C). We show that the set ∪A is independent. Otherwise there
exist elements {x0 , x1 , ..., xn } ∈ ∆ such that xi ∈ ∪A, for i = 0, ..., n.
Therefore there exists a set C ∈ A such that xi ∈ C for all i = 0, ..., n.
We conclude that C is dependent, C ∈ A, a contradiction. 158
4.
Ewa Graczyńska
Steinitz’ exchange theorem
A transfinite version of the Steinitz Exchange Theorem, provides that any
independent subset injects into any generating subset. For more information on the role of Steinitz papers consult the book chapter 400 Jahre Moderne Algebra, of [2]. The following is a generalization of Steinitz’ Theorem
originally proved in 1913 and then in [5]–[6]:
Theorem 18. If A is a basis and B is an independent subset (of a dependence space S). Then assuming Kuratowski-Zorn Maximum Principle,
there is a definite subset A0 of A such that the set B + (A − A0 ) is also a
basis of S.
Proof. If B is a basis then B is a maximal independent subset of S and
A0 = A is clear.
Assume that A is a basis and B is an independent subset (of the dependence space S). Consider X to be the family of all independent subsets
of S containing B and contained in A ∪ B. then (X, ⊆) is well ordered and
closed. Therefore assuming Kuratowski-Zorn Maximal Principle [7] there
exists a maximal element of X. We show that this maximal element X ∈ X
is a basis of S.
As X ∈ X then B ⊆ X ⊆ A ∪ B by the construction. Therefore
X = B + (A − A0 ) for some A0 ⊆ A. We show first that for all a ∈ A,
a ∼ ΣX. If a ∈ X then a ∼ ΣX by the definition. If not, then put
Y = X + {a}. Then X 6= Y , X ⊆ Y , B ⊆ Y ⊆ A ∪ B and Y is dependent
in S.
By the definition there exist elements: {x0 , x1 , ..., xn } ∈ ∆ with x1 , ...,
xn ∈ X + {a}, as X is an independent set. Moreover, one of xi is a, say
x0 = a. We get a ∼ ΣX as x0 , x1 , ..., xn are different.
Now we show that X is a basis of S. Let x ∈ S, then x ∼ ΣA as A
is a basis of S. Moreover for all a ∈ A, a ∼ ΣX, thus x ∼ ΣX by the
Transitivity Axiom. Theorem 19. If in a dependence space S with a dependence D, such that D
is transitive in the sense of [4] on all independent sets, then the transitivity
axiom of [5] holds.
Proof. Let x ∼ ΣA and a ∼ ΣB for all a ∈ A. Assume that x does
not depend of B, i.e. the set {x} ∪ B is independent. Therefore B is
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159
independent. But x ∈< A > and a ∈< B >, for all a ∈ A. Thus {a} ⊆<
B > and
S consequently < a >⊆<< B >>=< B >, for all a ∈ A. Therefore
A ⊆ {< a >: a ∈ A} ⊆< B >, and we get < A >⊆<< B >>=< B >.
Finally, x ∈< B >. Remark 20. The example presented in [4] on p. 425 shows that the assumption on transitivity of a dependence space S on independent sets is
an essential condition for the ”Exchange Lemma”. Namely, defining S =
{a, b, c} and D = {a, b}, {b, c}, {a, b, c} one gets: Ø, {a}, {b}, {c} and {a, c}
as independent sets. Moreover: < a >= {a, b} and << a >>= {a, b, c}.
Therefore the dependence D is not transitive on independent sets. The
bases are: {b} and {a, c}. The ”Exchange lemma” does not hold (for the
base A = {b} and the independent set B = {a}).
Remark 21. Let R denotes the set of real numbers. D is the set of all
infinite subsets of R. Then a dependence D in the sense [4] is given, which
is not a dependence space S in in the sense of [5].
Acknowledgement. The author expresses her thanks to the referee for
his valuable comments.
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[4] F. Gécseg, H. Jürgensen, Algebras with dimension, Algebra Universalis
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[6] N. J. S. Hughes, Steinitz’ Exchange Theorem for Infinite Bases II, Compositio Mathematica, tome 17 (1965–1966), pp. 152–155.
[7] K. Kuratowski, Une mthode d’élimination des nombres transfinis des raisonnements mathématiques, Fund. Math. 3 (1922), p. 89.
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Ewa Graczyńska
[8] K. Kuratowski, A. Mostowski, Teoria Mnogości, PWN, Warszawa 1966.
[9] D. J. A. Welsh, Matroid Theory, Academic Press, London, 1976.
Opole University of Technology
Institute of Matematics
e-mail: egracz@eranet.pl