A number of different problems of interest to the operational researcher and the mathematical economist - for example, certain problems of optimization on graphs and networks, of machine-scheduling, of convex analysis and of approx imation theory - can be formulated in a convenient way using the algebraic structure (R,$,@) where we may think of R as the (extended) real-number system with the binary combining operations x$y, x®y defined to be max(x,y),(x+y) respectively. The use of this algebraic structure gives these problems the character of problems of linear algebra, or linear operator theory. This fact hB.s been independently discovered by a number of people working in various fields and in different notations, and the starting-point for the present Lecture Notes was the writer's persuasion that the time had arrived to present a unified account of the algebra of linear transformations of spaces of n-tuples over (R,$,®),to demonstrate its relevance to operational research and to give solutions to the standard linear-algebraic problems which arise - e.g. the solution of linear equations exactly or approximately, the eigenvector eigenvalue problem andso on.Some of this material contains results of hitherto unpublished research carried out by the writer during the years 1970-1977.

Inhalt
1 Motivation.- 11 Introduction.- 12 Miscellaneous Examples.- 13 Conclusion: Our Present Aim.- 2 The Initial Axioms.- 21 Some Logical Geography.- 22 Commutative Bands.- 23 Isotone Functions.- 24 Belts.- 25 Belt Homomorphisms.- 26 Types of Belt.- 27 Dual Addition.- 28 Duality for Belts.- 29 Some Specific Cases.- 3 Opening and Closing.- 31 The Operations ??,???,??,???.- 32 The Principle of Closing.- 33 The Principle of Opening.- 4 The Principal Interpretation.- 41 Blogs.- 42 The Principal Interpretation.- 43 The 3-element Blog ?.- 44 Further Properties of Blogs.- 5 The Spaces En and ?mn.- 51 Band-Spaces.- 52 Two-Sided Spaces.- 53 Function Spaces.- 54 Matrix Algebra.- 55 The Identity Matrix.- 56 Matrix Transformations.- 57 Further Notions.- 6 Duality for Matrices.- 61 The Dual Operations.- 62 Some Matrix Inequalities.- 7 Conjugacy.- 71 Conjugacy for Belts.- 72 Conjugacy for Matrices.- 73 Two Examples.- 8 AA* Relations.- 81 Pre-residuation.- 82 Alternating AA* Products.- 83 Modified AA* Products.- 84 Some Bijections.- 85 A Worked Example.- 9 Some Schedule Algebra.- 91 Feasibility and Compatibility.- 92 The Float.- 93 A Worked Example.- 10 Residuation and Representation.- 101 Some Residuation Theory.- 102 Residuomorphisms.- 103 Representation Theorems.- 104 Representation for Matrices.- 105 Analogy with Hilbert Space.- 11 Trisections.- 111 The Demands of Reality.- 112 Trisections.- 113 Convex Subgroups.- 114 The Linear Case.- 115 Two Examples.- 12 ? ø Astic Matrices.- 121 ?ø Asticity.- 122 The Generalised Question 2.- 13 / Existence.- 131 / Existence Defined.- 132 CompatibleTrisections.- 133 Dually ? ø astic Matrices.- 134 / Defined Residuomorphisms.- 135 Omn As Operators.- 136 Some Questions Answered.- 14 The Equation A ? x = b Over a Bldg.- 141 Some Preliminaries.- 142 The Principal Solution.- 143 The Boolean Case.- 15 Linear Equations over a Linear Bldg.- 151 All Solutions of (143).- 152 Proving the Procedure.- 153 Existence and Uniqueness.- 154 A Linear Programming Criterion.- 155 Left-Right Variants.- 16 Linear Dependence.- 161 Linear Dependence Over El.- 162 The A Test.- 163 Some Dimensional Anomalies.- 164 Strong Linear Independence.- 17 Rank of Matrices.- 171 Regular Matrices.- 172 Matrix Rank Over A Linear Blog.- 173 Existence of Rank.- 18 Seminorms on En.- 181 Column-Spaces.- 182 Seminorms.- 183 Spaces of Bounded Seminorm.- 19 Some Matrix Spaces.- 191 Matrix Seminorms.- 192 Matrix Spaces.- 193 The Role of Conjugacy.- 20 The Zero-Lateness Problem.- 201 The Principal Solution.- 202 Case of Equality.- 203 Critical Paths.- 21 Projections.- 211 Congruence Classes.- 212 Operations in Rang A.- 213 Projection Matrices.- 22 Definite and Metric Matrices.- 221 Some Graph Theory.- 222 Definite Matrices.- 223 Metric Matrices.- 224 The Shortest Distance Matrix.- 23 Fundamental Eigenvectors.- 231 The Eigenproblem.- 232 Blocked Matrices.- 233 ø-Astic Definite Matrices.- 24 Aspects of the Eigenproblem.- 241 The Eigenspace.- 242 Directly Similar Matrices.- 243 Structure of the Eigenspace.- 25 Solving the Eigenproblem.- 251 The Parameter ?(A).- 252 Properties of ?(A).- 253 Necessary and Sufficient Conditions.- 254 The Computational Task.- 255 An Extended Example.- 26 Spectral Inequalities.-261 Preliminary Inequalities.- 262 Spectral Inequality.- 263 The Other Eigenproblems.- 264 More Spectral Inequalities.- 265 The Principal Interpretation.- 27 The Orbit.- 271 Increasing Matrices.- 272 The Orbit.- 273 The Orbital Matrix.- 274 A Practical Case.- 275 More General Situations.- 276 Permanents.- 28 Standard Matrices.- 281 Direct Similarity.- 282 Invertible Matrices.- 283 Equivalence of Matrices.- 284 Equivalence and Rank.- 285 Rank of ?.- 29 References and Notations.- 291 Previous Publications.- 292 Related References.- 293 List of Notations.- 294 List of Definitions.