By Andranick S. Tanguiane

Aggregation is the conjunction of knowledge, aimed toward its compact represen tation. Any time while the totality of information is defined when it comes to common ized symptoms, traditional counts, general representatives and attribute dependences, one without delay or not directly offers with aggregation. It contains revealing the main major features and exact positive aspects, quanti tative and qualitative research. consequently, the data turns into adaptable for additional processing and handy for human notion. Aggregation is generic in economics, information, administration, making plans, process research, and lots of different fields. this is why aggregation is so very important in info seasoned cessing. Aggregation of personal tastes is a selected case of the overall challenge of ag gregation. It arises in multicriteria decision-making and collective selection, while a collection of possible choices needs to be ordered with admire to contradicting standards, or a number of person evaluations. although, regardless of obvious similarity the issues of multicriteria decision-making and collective selection are a bit of various. certainly, an development in a few necessities on the expense of aggravate ing others isn't the similar because the delight of pursuits of a few contributors to the unfairness of the remainder. within the former case the reciprocal compensations are thought of inside of a definite entirety; within the latter we infringe upon the rights of self sufficient contributors. in addition, in multicriteria decision-making one usu best friend takes into consideration target components, while in collective selection one has to check subjective critiques which can't be measured properly.

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**Extra resources for Aggregation and Representation of Preferences: Introduction to Mathematical Theory of Democracy**

**Sample text**

2 Binary Relations and Orderings x x a o J 00000 00000 a I I 00000100000 b x x b a Fig. 24 a Fig. 25 indifferent to each other. This preference, shown in fig. 24, we shall denote (a, x .. ), where the comma separates the classes of indifferent alternatives, and x ... designates the class of elements indifferent to x. 10. EXAMPLE. If X consists of three elements at least, then for any two unequal a, b E X the relation P = {(a,x) : x E X, x =1= a} U {(b,x) : x E X, x =1= a,b} is a weak order, consequently, a preference with the element a being the best, b being the second best, and all the rest elements being inferior to a and band indifferent to each other.

Now we shall show that if P is a partial order on X and the induced indifference'" is transitive, then P is negatively transitive, whence P is a weak order. Let R be dual to P and (x,y) ¢: P, (y,z) ¢: P for some x,y,z E X, which implies (y,x) E R, (z,y) E R. 12 we have PeR, four cases are possible: (y,x)EP, (y,x) E R, (y,x)EP, (y,x) E R, (z,y)EP; (y,x) ¢: P, (z,y) E P; (z,y)ER, (z,y)¢:P; (y,x) ¢: P, (z,y) E R, (z,y) ¢: P. For the first case by transitivity we obtain (z,x) E P, whence by asymmetry (x, z) ¢: P.

PROPOSITION (Properties of Partial Orders). Let P be a partial order on X and R be dual to P. Then for arbitrary x, y, z E X it holds: 28 2 PREFERENCES AND GOAL FUNCTIONS 1. (x,y) E P implies (x,y) E R; 2. (x, y) E P and (y, z) E R implies (x, z) E R; 3. (x, y) E Rand (y, z) E P implies (x, z) E R; 4. (x, y) f/. P and (y, z) f/. R implies (x, z) E P; 5. (x, y) f/. Rand (y, z) f/. P implies (x, z) E P; 6. (x,y) f/. Rand (y,z) f/. R implies (x,z) E P; 7. (x,y) E P implies either (x,z) E P, or (z,y) E R; 8.