Join Geometries

1 The Join and Extension Operations in Euclidean Geometry.- 1.1 The Notion of Segment: Closed and Open.- 1.2 The Join of Two Distinct Points.- 1.3 Two Basic Properties of the Join Operation.- 1.4 A Crucial Question.- 1.5 The Join of Two Geometric Figures.- 1.6 Joins of Several Points: Does the Associative Law Hold?.- 1.7 The Join of Two Intersecting Geometric Figures.- 1.8 A Decision Must Be Made.- 1.9 The Join of a Point and Itself.- 1.10 The Unrestricted Applicability of the Join Operation.- 1.11 The Unrestricted Validity of the Associative Law for Join.- 1.12 The Universality of the Associative Law.- 1.13 Alternatives to the Definition aa = a.- 1.14 Convex Sets.- 1.15 A Geometric Proof in Join Terminology.- 1.16 A New Operation: The Idea of Extension.- 1.17 The Notion of Halfline or Ray.- 1.18 Formal Definition of the Extension Operation.- 1.19 Identification of Extension as a Geometrical Figure.- 1.20 Properties of the Extension Operation.- 1.21 The Extension of Two Geometric Figures.- 1.22 The Generation of Unbounded or Endless Figures.- 1.23 Is There an Alternative to Join as Open Segment?.- 2 The Abstract Theory of Join Operations.- 2.1 The Join Operation in Euclidean Geometry.- 2.2 Join Operations in a Set-Join Systems.- 2.3 The Postulates for the Operation Join.- 2.4 Application of the Theory to Euclidean Geometry.- 2.5 Elementary Properties of Join.- 2.6 Generalizations of J1-J4 to Sets.- 2.7 Extension of the Join Operation to n Terms.- 2.8 Comparison with the Conventional View of Geometry.- 2.9 Convex Sets.- 2.10 Joins, Intersections and Unions of Convex Sets.- 2.11 Interiors and Frontiers of Convex Sets.- 2.12 Euclidean Interiors and Frontiers.- 2.13 Interiority Properties of Convex Sets.- 2.14 Absorption by Joining.- 2.15 Closures of Euclidean Convex Sets, Intuitively Treated.- 2.16 The Closure of a Convex Set.- 2.17 Closure Properties of Convex Sets.- 2.18 Composition of the Interior and Closure Functions.- 2.19 The Boundary of a Convex Set.- 2.20 Project: Another Formulation of the Theory of Join.- 2.21 What Does the Theory Apply To?.- 2.22 The Triode Model.- 2.23 A Peculiarity of the Triode Model.- 2.24 The Cartesian Join Model.- 2.25 The Metric Interior of a Set in Euclidean Geometry.- 2.26 The Metric Closure of a Set in Euclidean Geometry.- 3 The Generation of Convex Sets-Convex Hulls.- 3.1 Introduction to Convexification: Two Euclidean Examples.- 3.2 Convexification of an Arbitrary Set.- 3.3 Finitely Generated Convex Sets-the Concept of Polytope.- 3.4 A Formula for a Polytope.- 3.5 A Distributive Law.- 3.6 An Absorption Property of Polytopes.- 3.7 Interiors and Closures of Polytopes.- 3.8 Powers of a Set and Polytopes.- 3.9 The Representation of Convex Hulls.- 3.10 Convex Hulls and Powers of a Set.- 3.11 Bounded Sets.- 3.12 Project-the Closed Join Operation.- 3.13 Convex Hulls of Finite Families of Sets-Generalized Polytopes.- 3.14 Properties of Generalized Polytopes.- 3.15 Examples of Generalized Polytopes.- 4 The Operation of Extension.- 4.1 Definition of the Extension Operation.- 4.2 The Extension Operation for Sets.- 4.3 The Monotonic Law for Extension.- 4.4 Distributive Laws for Extension.- 4.5 The Relation "Intersects" or "Meets".- 4.6 The Three Term Transposition Law.- 4.7 The Mixed Associative Law.- 4.8 Three New Postulates.- 4.9 Discussion of the Postulates.- 4.10 Formal Consequences of the Postulate Set J1-J7.- 4.11 Formal Consequences Continued.- 4.12 Joins and Extensions of Rays.- 4.13 Solving Problems.- 4.14 Extreme Points of Convex Sets.- 4.15 Open Convex Sets.- 4.16 The Intersection of Open Sets.- 4.17 The Join of Open Sets.- 4.18 Segments Are Infinite.- 4.19 The Polytope Interior Theorem.- 4.20 The Interior of a Join of Convex Sets.- 4.21 The Generalized Polytope Interior Theorem.- 4.22 Closed Convex Sets.- 4.23 The Theory of Order.- 4.24 Ordered Sets of Points.- 4.25 Separation of Points by a Point.- 4.26 Perspectivity and Precedence of Points.- 5 Join Geometries.- 5.1 The Concept of a Join Geometry.- 5.2 A List of Join Geometries.- 5.3 Deducibility and Counterexamples.- 5.4 The Existence of Points.- 5.5 Isomorphism of Join Systems.- 5.6 A Class of Join Geometries of Arbitrary Dimension.- 5.7 ?n Is Converted into a Vector Space.- 5.8 Restatement of the Definition of Join in ?n.- 5.9 Proof That (?n, *) Is a Join Geometry.- 5.10 Linear Inequalities and Halfspaces.- 5.11 Pathological Convex Sets.- 5.12 An Infinite Dimensional Join Geometry.- 5.13 Three Pathological Convex Sets.- 5.14 Is There a Simple Way to Construct Pathological Convex Sets?.- 6 Linear Sets.- 6.1 The Notion of Linear Set.- 6.2 The Definition of Linear Set.- 6.3 Conditions for Linearity.- 6.4 Constructing Linear Sets from Linear Sets.- 6.5 The Construction of a Linear Set from a Convex Set.- 6.6 Linear Sets Give Rise to Join Geometries.- 6.7 The Generation of Linear Sets: Two Euclidean Examples.- 6.8 The Generation of Linear Sets: General Case.- 6.9 The Linear Hull of a Finite Set: Finitely Generated Linear Sets.- 6.10 The Definition of Line.- 6.11 The Linear Hull of an Arbitrary Set.- 6.12 The Linear Hull of a Finite Family of Sets.- 6.13 Linear Hulls of Interiors and Closures.- 6.14 Geometric Relations of Points-Linear Dependence and Independence.- 6.15 Properties of Linearly Independent Points.- 6.16 Simplexes.- 6.17 Linear Dependence and Intersection of Joins.- 6.18 Covering in the Family of Linear Sets-Hyperplanes.- 6.19 The Height of a Linear Set-Approach to a Theory of Dimension.- 6.20 Linear Sets and the Interior Operation.- 6.21 Applications: Interiority Properties.- 6.22 The Prevalence of Nonpathological Convex Sets.- 6.23 The Linear Hull of a Pair of Convex Sets.- 6.24 The Interior of the Extension of Two Convex Sets.- 7 Extremal Structure of Convex Sets: Components and Faces.- 7.1 The Notion of an Extreme Set of a Convex Set.- 7.2 The Definition of Extreme Set of a Convex Set.- 7.3 Remarks on the Definition.- 7.4 Elementary Properties of Extreme Sets.- 7.5 The Interior Operation Is Applied to Extreme Sets.- 7.6 The Closure Operation Is Applied to Extreme Sets.- 7.7 Other Characterizations of Extremeness.- 7.8 Is Extremeness Preserved under Join and Extension?.- 7.9 Extreme Sets with a Preassigned Interior Point.- 7.10 Classifying Extreme Sets.- 7.11 Open Extreme Sets: Components.- 7.12 The Partition Theorem for Convex Sets.- 7.13 The Component Structure of a Convex Set.- 7.14 Components as Maximal Open Subsets.- 7.15 Intersection Properties of Components.- 7.16 Components and Perspectivity of Points.- 7.17 Components of Polytopes.- 7.18 The Concept of a Face of a Convex Set.- 7.19 The Nonseparation Property of a Face.- 7.20 Elementary Properties of Faces of Convex Sets.- 7.21 Additional Properties of Faces.- 7.22 Facial Structure of Convex Sets.- 7.23 Facial Structure Continued.- 7.24 A Correspondence Between Components and Faces.- 7.25 Faces of Polytopes.- 7.26 Covering in the Family of Faces of a Convex Set.- 7.27 Extreme Sets and Extremal Linear Spaces.- 7.28 Associated Extreme Sets.- 7.29 Covering of Extremals Arising from Faces.- 7.30 Extremal Hyperplanes and Exposed Faces.- 7.31 Supporting and Tangent Hyperplanes.- 8 Rays and Halfspaces.- 8.1 Elementary Properties of Rays.- 8.2 Elementary Operations on Rays.- 8.3 Opposite Rays.- 8.4 Separation of Two Rays by a Common Endpoint.- 8.5 The Partition of Space into Rays.- 8.6 Closed Rays.- 8.7 The Linear Hull of a Ray.- 8.8 The Halfspaces of a Linear Set.- 8.9 A Point of Terminology.- 8.10 Elementary Properties of L-Rays.- 8.11 Elementary Operations on L-Rays.- 8.12 Opposite Halfspaces.- 8.13 Separation of Two Halfspaces by a Common Edge.- 8.14 The Partition of Space into Halfspaces.- 8.15 Closed Halfspaces.- 8.16 The Linear Hull of a Halfspace.- 9 Cones and Hypercones.- 9.1 Cones.- 9.2 Convex Cones.- 9.3 Pointed Convex Cones.- 9.4 Generation of Convex Cones.- 9.5 How Shall Convex o-Cones Be Generated?.- 9.6 Polyhedral Cones.- 9.7 Finiteness of Generation of Convex Cones.- 9.8 Extreme Rays.- 9.9 Extreme Rays of Convex Cones.- 9.10 The Analogous Development Breaks Down.- 9.11 Regularly Imbedded Rays.- 9.12 The Generation of Convex Cones by Extreme Rays.- 9.13 Hypercones.- 9.14 Elementary Properties of Hypercones.- 9.15 Convex Hypercones.- 9.16 Tapered Convex Hypercones.- 9.17 Generation of Convex Hypercones.- 9.18 Polyhedral Hypercones.- 9.19 Finiteness of Generation of Convex Hypercones.- 9.20 Extreme Halfspaces of Convex Hypercones.- 9.21 Regularly Imbedded Half spaces.- 9.22 The Generation of Convex Hypercones by Extreme Half spaces.- 10 Factor Geometries and Congruence Relations.- 10.1 Congruence Relations Determined by Halfspaces.- 10.2 Pairs of Congruences.- 10.3 Congruence Relations between Sets.- 10.4 Properties of Modular Set Congruence.- 10.5 Families of Halfspaces.- 10.6 New Notations.- 10.7 A Join Operation in J : M.- 10.8 Factor Geometries.- 10.9 J: M Satisfies J1-J6.- 10.10 Geometric Interpretation of Factor Geometries.- 10.11 Continuation of the Theory of Factor Geometries.- 10.12 Differences Between J : M and J.- 10.13 A Correspondence Relating J : M and J.- 10.14 Convexity in a Factor Geometry.- 10.15 Linearity in a Factor Geometry.- 10.16 The Cross Section Correspondence.- 10.17 Isomorphism of Factor Geometries.- 11 Exchange Join Geometries-The Theory of Incidence and Dimension.- 11.1 Exchange Join Geometries.- 11.2 Two Points Determine a Line.- 11.3 A Basis for a Linear Space.- 11.4 Incidence Properties.- 11.5 The Exchange Theorem.- 11.6 Rank and Dimension.- 11.7 Rank and Linear Containment.- 11.8 Rank and Covering.- 11.9 The Dimension Principle.- 11.10 Characterizations of Linear Dependence.- 11.11 Rank of a Set of Points.- 11.12 Rank and Height of Linear Spaces.- 12 Ordered Join Geometries.- 12.1 Ordered Join Geometries.- 12.2 Two Points Determine a Line.- 12.3 A New Formula for Line ab.- 12.4 A Formula for a Ray.- 12.5 Another Formula for a Line.- 12.6 An Expansion Formula for a Segment.- 12.7 An Expansion Formula for a Join of Points.- 12.8 An Expansion for a Polytope.- 12.9 Modularity Principles.- 12.10 The Section of a Polytope by a Linear Space.- 12.11 Consequences of the Polytope Section Theorem.- 12.12 Polytopes Are Closed.- 12.13 The Interior of a Polytope-Another View.- 12.14 Separation Property of the Frontier of a Polytope.- 12.15 A New Formula for Linear Hulls.- 12.16 Convex Dependence and Independence.- 12.17 How Are C-Dependence and L-Dependence Related?.- 12.18 Convex Dependence and Independence in an Ordered Geometry.- 12.19 Convex Dependence and Polytopes.- 12.20 Helly's Theorem.- 12.21 Decomposition of Polytopes into Subpolytopes: Caratheodory's Theorem.- 12.22 Equivalence of Linear and Convex Independence.- 12.23 The Separation of Linear Spaces by Linear Spaces.- 13 The Structure of Polytopes in an Ordered Geometry.- 13.1 Enclosing Convex Sets in Closed Halfspaces.- 13.2 Interpreting the Results.- 13.3 Supporting Halfspaces to Convex Sets.- 13.4 Supporting Hyperplanes to Polytopes Exist.- 13.5 Polytopes Are Intersections of Closed Half spaces.- 13.6 Covering of Faces of a Polytope and Their Linear Hulls.- 13.7 Facets of a Polytope.- 13.8 Facial Structure of a Polytope.- 13.9 Facets of a Facet of a Polytope.- 13.10 Generation of Convex Sets by Extreme Points.- 13.11 Terminal Segments and Dispensability of Generating Points.- 13.12 Dispensability Sequences.- 13.13 Generation Conditions.- 13.14 Segmental Closure and Linear Boundedness.- 13.15 Convex Sets Generated by Their Extreme Points.- 13.16 A Characterization of a Polytope.- 13.17 Polytopes and Intersections of Closed Halfspaces.- References.