By I.M. Yaglom, I.G. Volosova

The current booklet relies at the lecture given via the writer to senior scholars in Moscow at the twentieth of April of 1966. the excellence among the cloth of the lecture and that of the booklet is that the latter comprises workouts on the finish of every part (the so much tricky difficulties within the workouts are marked via an asterisk). on the finish of the e-book are positioned solutions and tricks to a couple of the issues. The reader is suggested to resolve many of the difficulties, if now not all, simply because purely after the issues were solved can the reader make certain he is aware the subject material of the ebook. The e-book includes a few not obligatory fabric (in specific, Sec. 7 and Appendix that are starred within the desk of contents) that may be passed over within the first interpreting of the publication. The corresponding components of the textual content of the e-book are marked through one big name at first and by way of stars on the finish. even if, within the moment interpreting of the publication you should research Sec. 7 because it includes a few fabric very important for sensible purposes of the speculation of Boolean algebras.

The bibliography given on the finish of the booklet lists a few books which are of use to the readers who are looking to examine the speculation of Boolean algebras extra thoroughly.

The writer is thankful to S. G. Gindikin for worthwhile recommendation and to F. I. Kizner for the thoroughness and initiative in enhancing the ebook.

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Indeed, for any number x satis( х ® у ; ф z=(x®zmy@z) x®z = y®z x®y oi—o-!. у х о Z ii oi—о (a) о X (b) Fig. 14 30 Z (x®y)®z=(x®z)®(y®z) x®y=x®z У®г о y II the condition 0 ^ x ^ 1 we always have x © 0 = max [x, 0] = x and x ® 1 = min [x, 1] = x x Ф 1 — max [x, 1] = 1 and x ® 0 = min [x, 0] = 0 lying Example 4. Algebra of least common multiples and greatest common divisors. Let N be an arbitrary integer. As the elements of the new algebra we shall take all the possible divisors of the number N. For instance, if N = 210 = = 2 -3 -5 -7 then the elements of the algebra in question are the numbers 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105 and 210.

Similarly, the negation of the proposition "the pupil has no bad marks" means thai "the pupil has bad^marks" and the double negation of the former proposition slates that "it is falsi' llial the pupil has bad marks" and is therefore equivalent to the original proposition asserting t h a t the pupil has no bad marks. The De Morgan rules a-\-b — ab ab-=a-\-b and for the propositions are also very i m p o r t a n t ; the verbal s t a t e m e n t of lliese rules is a little more complicated (in S h is connection see Exercise 1 below).

A) Let N = ргр2 . . , Ph a r e pairwise different. Prove t h a t in this case the "algebra of least common multiples and greatest common divisors" whose elements are the divisors of the number N (see Example 4 on page 31) reduces to the "algebra of the subsets of the universal set / = p2, . , ph}". Proceeding from this fact show that in this "algebra of least common multiples and greatest common divisors" all the laws of a Boolean algebra hold including the De Morgan rules. (b) Let TV = pA where p is a prime number and A is a positive integer.

### An Unusual Algebra by I.M. Yaglom, I.G. Volosova

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