By Andranick S. Tanguiane

ISBN-10: 3642765165

ISBN-13: 9783642765162

ISBN-10: 3642765181

ISBN-13: 9783642765186

Aggregation is the conjunction of data, aimed toward its compact represen tation. Any time while the totality of information is defined by way of common ized signs, traditional counts, general representatives and attribute dependences, one at once or not directly offers with aggregation. It contains revealing the main major features and particular beneficial properties, quanti tative and qualitative research. for this reason, the knowledge turns into adaptable for extra processing and handy for human notion. Aggregation is ordinary in economics, facts, administration, making plans, method research, and lots of different fields. this is why aggregation is so vital in info seasoned cessing. Aggregation of personal tastes is a specific case of the overall challenge of ag gregation. It arises in multicriteria decision-making and collective selection, while a collection of choices needs to be ordered with recognize to contradicting standards, or a number of person evaluations. in spite of the fact that, even with obvious similarity the issues of multicriteria decision-making and collective selection are a bit of diversified. certainly, an development in a few requisites on the fee of irritate ing others isn't the similar because the delight of pursuits of a few members 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 autonomous members. furthermore, in multicriteria decision-making one usu best friend takes into consideration target components, while in collective selection one has to match subjective critiques which can't be measured properly.

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SI is symmetric if and only if S2 is symmetric; 3. SI is asymmetric if and only if S2 is connected; 4. SI is anti-symmetric if and only if S2 is weakly-connected; 5. SI is transitive if and only if S2 is negatively transitive. We omit the trivial proof of this proposition. Now we define three types of binary relations, to which we refer in our study as to preferences. 4. DEFINITION. A partial order is defined to be an asymmetric transitive binary relation. 5. DEFINITION. A weak order is defined to be an asymmetric negatively transitive binary relation.

Consider the two arrows space-the strictly ordered topological space, consisting of continuum of pairs of points, ordered lexicographically. In other words, let space X consist of the couples (x, y), where x 50 2 PREFERENCES AND GOAL FUNCTIONS (x,l) I "( A base neighborhood of the point (x,O) A base neighborhood of the point (x, 1) "( "( )" (x,O) Fig" 35 runs through the segment [0; 1] and y adopts the values 0, or 1, ordered by the rule: (x', y') >- (x", y") if either x' > x", or x' = x" and y' > y".

THEOREM (About the Existence of a Goal Function). Let X be a strict, or in dual terms linear ordered set, regarded as a topological space with the topology induced by the given strict order r-. Then the following conditions are equivalent: 1. There exists a goal function on X. 2. There exists a monotone homeomorphism from X into the interval (0; 1). 3. The topology on X induced by the strict order has a countable base. 4. X as a topological space is separable and has no more than a countable set of jumps (empty intervals).

### Aggregation and Representation of Preferences: Introduction to Mathematical Theory of Democracy by Andranick S. Tanguiane

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