Relational MathematicsRelational mathematics is to operations research and informatics what numerical mathematics is to engineering: it is intended to help modelling, reasoning, and computing. Its applications are therefore diverse, ranging from psychology, linguistics, decision aid, and ranking to machine learning and spatial reasoning. Although many developments have been made in recent years, they have rarely been shared amongst this broad community of researchers. This first comprehensive overview begins with an easy introduction to the topic, assuming a minimum of prerequisites; but it is nevertheless theoretically sound and up to date. It is suitable for applied scientists, explaining all the necessary mathematics from scratch using a multitude of visualised examples, via matrices and graphs. It ends with tangible results on the research level. The author illustrates the theory and demonstrates practical tasks in operations research, social sciences and the humanities. |
Contents
1 Introduction | 1 |
PART I REPRESENTATIONS OF RELATIONS | 3 |
2 Sets subsets and elements | 5 |
3 Relations | 15 |
PART II OPERATIONS AND CONSTRUCTIONS | 33 |
4 Algebraic operations on relations | 35 |
the standard view | 49 |
6 Relations and vectors | 91 |
13 Preference and indifference | 349 |
14 Aggregating preferences | 368 |
15 Relational graph theory | 396 |
16 Standard Galois mechanisms | 415 |
PART V ADVANCED TOPICS | 439 |
17 Mathematical applications | 441 |
18 Implication structures | 461 |
19 Power operations | 483 |
7 Domain construction | 106 |
PART III ALGEBRA | 155 |
8 Relation algebra | 157 |
9 Orders and lattices | 183 |
10 Rectangles fringes inverses | 200 |
11 Concept analysis | 251 |
PART IV APPLICATIONS | 301 |
an advanced view | 303 |
Common terms and phrases
according to Prop Alfred Tarski algebraic antichains applied Arbuthnot arrow assume baseset bijective bisimulation Boolean Botticelli closure complete lattice concept consider construct defined definition diagonal difunctional relation direct sum Draw Dupont element example exist existential image Ferrers relation finite formulae fringe fringe(R given graph Hasse diagram homogeneous relation homomorphism injective intersection interval graph intervalorder irreflexive lattice linear order linear strictorder Loss mapping mathematics matrix maxclique membership relation natural projection negation non-enlargeable rectangle numbers obtain pair of sets Perez permutation point-free possibly heterogeneous powerset preorder Proof properties Proposition prove quotient set rearranged reflexive relation algebra result rows and columns satisfied semiorder Spanish subset surjective symmetric quotient Theorem transposed trivial univalent upper bound vector visualized weakorder ετ ηψ ㅇㅇ