Classes and theories of trees associated with a class of linear orders

- Goranko, Valentin, Kellerman, Ruaan

**Authors:**Goranko, Valentin , Kellerman, Ruaan**Date:**2010-08**Subjects:**Linear orders , First-order theories , Nondefinable paths , Trees (Graph theory)**Type:**Article**Identifier:**http://ujcontent.uj.ac.za8080/10210/364640 , uj:5832 , ISSN 1367-0751 , http://hdl.handle.net/10210/7849**Description:**Given a class of linear order types C, we identify and study several different classes of trees, naturally associated with C in terms of how the paths in those trees are related to the order types belonging to C. We investigate and completely determine the set-theoretic relationships between these classes of trees and between their corresponding first-order theories. We then obtain some general results about the axiomatization of the first-order theories of some of these classes of trees in terms of the first-order theory of the generating class C, and indicate the problems obstructing such general results for the other classes. These problems arise from the possible existence of nondefinable paths in trees, that need not satisfy the first-order theory of C, so we have started analyzing first-order definable and undefinable paths in trees.**Full Text:**

**Authors:**Goranko, Valentin , Kellerman, Ruaan**Date:**2010-08**Subjects:**Linear orders , First-order theories , Nondefinable paths , Trees (Graph theory)**Type:**Article**Identifier:**http://ujcontent.uj.ac.za8080/10210/364640 , uj:5832 , ISSN 1367-0751 , http://hdl.handle.net/10210/7849**Description:**Given a class of linear order types C, we identify and study several different classes of trees, naturally associated with C in terms of how the paths in those trees are related to the order types belonging to C. We investigate and completely determine the set-theoretic relationships between these classes of trees and between their corresponding first-order theories. We then obtain some general results about the axiomatization of the first-order theories of some of these classes of trees in terms of the first-order theory of the generating class C, and indicate the problems obstructing such general results for the other classes. These problems arise from the possible existence of nondefinable paths in trees, that need not satisfy the first-order theory of C, so we have started analyzing first-order definable and undefinable paths in trees.**Full Text:**

Logical theories of orthogonality structures

**Authors:**Kellerman, Ruaan**Date:**2012-02-28**Subjects:**Orthogonal functions**Type:**Mini-Dissertation**Identifier:**uj:2080 , http://hdl.handle.net/10210/4427**Description:**M.Sc. , The geometric relation of orthogonality - when the angle determined by two lines is a right angle - has a rich and interesting theory. We investigate geometric orthogonality structures from a formal logical and model theoretic viewpoint, both using a domain consisting of lines, where the orthogonality relation is a binary line relation, and also using a domain consisting of points, where pairs of distinct points are treated as lines and the orthogonality relation is treated as a quaternary relation on points. We establish first-order axiom systems for these orthogonality structures for dimension n 2 2, and examine some of the metamathematical properties of these structures and their associated theories.**Full Text:**

**Authors:**Kellerman, Ruaan**Date:**2012-02-28**Subjects:**Orthogonal functions**Type:**Mini-Dissertation**Identifier:**uj:2080 , http://hdl.handle.net/10210/4427**Description:**M.Sc. , The geometric relation of orthogonality - when the angle determined by two lines is a right angle - has a rich and interesting theory. We investigate geometric orthogonality structures from a formal logical and model theoretic viewpoint, both using a domain consisting of lines, where the orthogonality relation is a binary line relation, and also using a domain consisting of points, where pairs of distinct points are treated as lines and the orthogonality relation is treated as a quaternary relation on points. We establish first-order axiom systems for these orthogonality structures for dimension n 2 2, and examine some of the metamathematical properties of these structures and their associated theories.**Full Text:**

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