The Collinearity Structure

Summary. The text includes basic axioms and theorems concerning the collinearity structure based on Wanda Szmielew [1], pp. 18-20. Collinearity is defined as a relation on Cartesian product [:S, S, S:] of set S. The basic text is preceeded with a few auxiliary theorems (e.g: ternary relation). Then come the two basic axioms of the collinearity structure: A1.1.1 and A1.1.2 and a few theorems. Another axiom: Aks dim, which states that there exist at least 3 non-collinear points, excludes the trivial structures ( i.e. pairs 〈S,[:S, S, S:] 〉). Following it the notion of a line is included and several additional theorems are appended. MML Identifier: COLLSP. The articles [3], and [2] provide the notation and terminology for this paper. In the sequel R, X will denote sets. Let us consider X. A set is said to be a ternary relation on X if: it ⊆ [: X, X, X:]. Next we state two propositions: (1) R is a ternary relation on X if and only if R ⊆ [:X, X, X:]. (2) X = ∅ or there exists arbitrary a such that {a} = X or there exist arbitrary a, b such that a � = b and a ∈ X and b ∈ X. We consider collinearity structures which are systems 〈points, a collinearity relation〉, where the points constitute a non-empty set and the collinearity relation is a ternary relation on the points. In the sequel CS is a collinearity structure. Let us consider CS. A point of CS is an element of the points of CS. In the sequel a, b, c denote points of CS. Let us consider CS, a, b, c. We say that a, b and c are collinear if and only if: 〈a,b,c 〉 ∈ the collinearity relation of CS. The following proposition is tru