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 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. 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 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 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 2 ∆. Definition 3. An element x is called dependent on A and is denoted by x ∼ ΣA if either x 2 A or if there exist distinct elements x0; x1; :::; xn such that (1) fx0; x1; :::; xng 2 ∆ where x0 = x and x1; :::; xn 2 A and directly dependent on fxg or fx0; x1; :::; xng, respectively. Definition 4. A set A is called dependent if (1) is satisfied for some distinct elements x0; x1; :::; xn 2 A, and otherwise A is independent. Definition 5. If a set A is independent and for any x 2 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 2 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 2 S, A ⊆ S. The element x is said to depend on A if a 2 A or there is an independent subset A0 ⊆ A such that A0 [ fag is dependent in S. Dependence Spaces 155 Lemma 8. In a dependence space S the following are equivalent for an element x 2 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 2 A or there exists elements x0 = x and x1; :::; xn 2 A such that fx0; x1; :::; xng 2 ∆. If the set fx1; :::; xng is dependent in S in the sense 0 of [5], then there exists its finite subset, say A ⊆ fx1; :::; xng, such that A0 2 ∆, 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 fxg [ 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 2 A or there exists an independent subset A0 ⊆ A such that A0 [fxg is dependent in S. In the second case we get that fxg [ A0 is dependent in 0 S, i.e. there exists x0 = x and x1; :::; xn 2 A , such that x0; x1; :::; xn 2 ∆, 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 2< 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 2<< X >>, i.e. x ∼ Σ < X >. But y ∼ ΣX, for all y 2< X >. Thus by the Transitivity Axiom x ∼ ΣX, i.e. x 2< 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 2 A, a ∼ ΣB. Thus x 2 A or the set fxg [ A is dependent, i.e. x 2< A >. Moreover, a 2< B >, for all a 2 A. Therefore fag ⊆< B >, i.e. < a >⊆<< B >>=< B >, by the Remark 3.6 of [4]. In consequence: A = Sfa : a 2 Ag ⊆ Sf< a >: a 2 Ag ⊆< B > and thus: < A >⊆<< B >>=< B >. We get that x 2< 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 f0g 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.