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By Redei L.

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According to our definition, the terms may be any, not necessarily distinct, objects. 6) and a further sequence bI.... , n). 6) n is called the number of the terms of the sequence; (8;6) itself is also called a sequence of n terms. Further ai is called the i`h term of the sequence, i itself the index of this term. and call it an (ordered) n-tuple. , In the special cases n = 2, 3, 4 the terms pair (pair of elements), triple, and quadruple are also used. ) If the pairs (a, b), (b, a) are not to be distinguished, we speak of a non-ordered pair (a, b), etc.

If O(S) = 1 we may call S a structure with one element. If, moreover, S is a group, a module or a ring, then the single element of S must necessarily be the neutral element of S. ) Accordingly, we call the structure with one element in the three cases: unity group, null module and null ring. We do not distinguish them from their unique element. ) Obviously, with the unity group, the null module and the null ring, we have enumerated all structures which consist of only one element. We often leave these trivial structures out of consideration, but in some cases they are important.

As regards multiplication of integers, the number 1 is obviously the unity element for which we retain the normal notation. We shall sometimes denote the unity element of a set by 1 even when it is not the number I provided that it will give rise to no misunderstanding. If there is such a possibility then we may use the symbol 1. > of even numbers, let the customary multiplication be defined. ) Since for even numbers a, b (0 0) ab ¢ b, there is no unity element in this set. In the set of mappings of a set into itself, obviously the identical mapping is the unity element.

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Algebra Vol. I by Redei L.


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