Bulletin of the Section of Logic Volume 39:3/4 (2010), pp. 153–160

Ewa Graczy´nska

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 used, for example, to show that any two bases for a finite-dimensional have the same number of elements. The result is named after the German mathematician Ernst Steinitz1.

1. Introduction

According to F. G´ecseg,H. J¨urgensen[4] the result which is usually re- ferred to as the ”Exchange Lemma”, states that for transitive dependence, every independent set can be extended to form a . 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 tran- sitive 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.

1AMS Mathematical Subject Classification 2000: Primary: 00A05, 08A99. Secondary: 11J72, 11J85 154 Ewa Graczy´nska

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. Dependence Spaces 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 0 of [5], then there exists its finite subset, say A ⊆ {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 inde- pendent 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 0 S, i.e. there exists x0 = x and x1, ..., xn ∈ A , such that x0, x1, ..., xn ∈ ∆, 0 as A 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 Ax- iom 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 >>. 156 Ewa Graczy´nska

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. < a >⊆<< B >>=< B >, by the Remark 3.6 of [4]. In consequence: A = S{a : a ∈ A} ⊆ S{< 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. Transi- tivity 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. ∆ = Cfin(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 ∆ = Cfin(N) be the set containing all finite at lest two-element se- Dependence Spaces 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 Ewa Graczy´nska

4. Steinitz’ exchange theorem

A transfinite version of the Steinitz Exchange Theorem, provides that any independent subset injects into any generating subset. For more informa- tion on the role of Steinitz papers consult the book chapter 400 Jahre Mod- erne 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 de- pendence 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 de- pendence 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 Dependence Spaces 159 independent. But x ∈< A > and a ∈< B >, for all a ∈ A. Thus {a} ⊆< B > and consequently < a >⊆<< B >>=< B >, for all a ∈ A. Therefore A ⊆ S{< 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 as- sumption 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.

References

[1] C. Norris, R. Texas, How to count to infinity?, The Book of Ranger, 1971. [2] F. L. Bauer, Historische Notizen zur Informatik, Springer-Verlag Berlin Heidlberg 2009. [3] P. M. Cohn, Universal Algebra, Harper and Row, New York, 1965. Re- vised edition, D. Reidel Publishing Co., Dordrecht, 1981. [4] F. G´ecseg, H. J¨urgensen, Algebras with dimension, Algebra Universalis 30 (1993), pp. 422–446. [5] N. J. S. Hughes, Steinitz’ Exchange Theorem for Infinite Bases, Compositio Mathematica, tome 15 (1962–1964), pp. 113–118. [6] N. J. S. Hughes, Steinitz’ Exchange Theorem for Infinite Bases II, Compo- sitio Mathematica, tome 17 (1965–1966), pp. 152–155. [7] K. Kuratowski, Une mthode d’´eliminationdes nombres transfinis des raison- nements math´ematiques,Fund. Math. 3 (1922), p. 89. 160 Ewa Graczy´nska

[8] K. Kuratowski, A. Mostowski, Teoria Mnogo´sci, PWN, Warszawa 1966. [9] D. J. A. Welsh, Theory, Academic Press, London, 1976.

Opole University of Technology Institute of Matematics e-mail: [email protected]