Download PDF by Redei L.: Algebra Vol. I

By Redei L.

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We now want to interpret an infinite sequence a,, a2, ... or ao, ai, ... , a_, ao, a,, ... 6). , b there arises, by juxtaposition, the sequence al, . , b,. 8) are also known as countably infinite sequences as their indices constitute a countably infinite set. Finite and countably infinite sequences are countable sequences. When speaking of a sequence al, a2, ... or set may often be understood, provided that there is no possibility of misinterpretation. A non-ordered pair (a, b) is more general than the set

We shall generalize these and similar well-known examples below. >, if an element of S is uniquely assigned to each ordered pair a, j9 of elements of S in a certain way. We denote this element by a 0 j3 where 0 is the sign of composition. This may then be called the "composition 0". Alternatively, the composition is a mapping of the product set S X S into S or a functionfloc, fi) defined in S. However, since the above concept of compositions is too general, we shall only consider in detail compositions with certain special properties.

By an extension of the concept of a sequence, we take this to mean a mapping of any ordered set 1= < . > into an arbitrary set. , ... <,U< ... I itself is called the index set (of this sequence). In a term aK, K is called the index of this term. If I has a minimal element (as is usually the case), the sequence becomes a2, a#, ... , where 1, 2, ... no longer denote the natural numbers, but are the elements of an arbitrary index set. ) sets with arbitrary cardinal number, not only the countable ones.

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Algebra Vol. I by Redei L.

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