# Banach algebras: An introduction (Pure and applied by Ronald Larsen By Ronald Larsen

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Asymmetrical relations arer we may Kinds of Relations +s say, the most characteristically relational of relations, and the most important to the philosopher who wishes to study the ultimate logical nature of relations. e, relations which at most one term can have to a given term. Such are father, mother, husband (except in Tibet), square of, sine of, and so on. But parent, squareroot, and so on, are not one-many. It is possible, formally, to replace all relations by one-many relations by means of a device.

The structure of the maP corresPondawith that of 52 Similaritl of Relations j3 the country of which it is a map. The space-relationsin the map have " likeness" to the space-relations in the country mapped. It is this kind of connection between relations that we wish to define. We *"y, in the first place, profitably introduce a certain restriction. e. to such as permit of the formation of a single class out of the domain and the converse domain. This is not always the case. e. the relation which the domain of a relation has to the relation.

Z) If & is betweena and * and alsobetweena andy, then either x and y arc identical, or tc is between b and lt or y is between b and x. edin the caseof points on a straight line in ordinary space. Aoy three-term relation which verifies them gives rise to series,as may be seenfrom the following definitions. For the sake of definiteness,let us assume I Cl. Riadstad,i Matematiaa, iv. pp. ; Pri'nciptresof Mathemat'ics, p. 3e4 (\$ 375). b. Then the points of the line (ab) arc Q) those between which and b, a lies-these we will call to the left o f .

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