Cosets in a Semi-Group

Abstract

INTRODUCTION. If two integers a and b have the property, a b is divisible by the integer m, a is said to be congruent to b modulo m. This is written a b (mod m). The equality of ordinary algebra shares many of its properties with the congruence relation. In this paper, we shall suppose that the reader is familiar with this relation as discussed in the books on elementary numrber theory. The congruence characteristics of the integers which are prime to the modulus are well known. These integers will be called units in this study. One of the main objectives of this article is to present some of the properties of the integers which are not prime to the modulus. 1 Since no two of the integers 0, 1, . . ., m 1 are congruent modulo m, and since every integer is congruent to one of them, they are called the least residues modulo m. The formula i = km + r, 0 < r < m, gives us a method of finding the least residue of an integer i modulo m. We shall now give some examples illustrating the behavior of certain integers not prime to a composite modulus. When 30 is raised to successive powers, we obtain the set of incongruent integers modulo 360: 30, 180, 0. Similarly 12 generates the least residues: 12, 144, 288, 216, 72 modulo 360. This set contains the set: 144, 288, 216, 72, which has the properties modulo 360: each one divides the others modulo 360; a 216 a (mod 360) for each a of the subset; and 288 generates the subset. 5 generates a set of least residues modulo 360, which has the property: each divides the others modulo 360. The least residues of the units modulo 12 are: 1, 5, 7, 11. If the elements in this set are multiplied successively by 1, 2, 3, 4, 6, and 0, we get the sets of least residues mo3ulo 12: 1, 5, 7, 11; 2, 10; 3, 9; 4, 8; 6; and 0. Furthermore these sets are disjoint and they exhaust the total set of least residues modulo 12. These examples are special cases of theorems 1, 3, and 4. Cosets in semi-groups are mentioned in theorem 1; we shall develop a theory of these cosets which is similar to the theory of cosets in groups. It is astounding that theorems 3, 4, and 6 concerning the properties of the residues modulo m are not well known; however the writer has not beeni able to find published proofs of these theorems. If any previous work is duplicated, then that part of the present paper can be regarded as exposi tory.