By N. Hitchin

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N a 2 = M. Show that the monoid M is monoid isomorphic with the Cartesian product monoid M°l X M"2 [cf. 8)]. 4H. For relations a, ß, y on a set A, show that (a n /3)° y ç ( a ° y ) n (ß°y). Give an example where the inclusion is proper. 41. For relations a, ß,y on a set ^4, show that ( a U /3)° y = ( a ° y) U ( 0 » y ). 4J. Identify Z w with n ^ , , ^ / , , via m<60> - (m<3>, m<4>, m<5>). Interpret n 3 s „ s 5Z„ (cf. 2Q) as the set of sections s : {3,4,5} -» E 3 s „ s 5 / „ of the disjoint union p : E 3 s „ á 5 Z n -» {3,4,5} of the maps pn : Z„ -» {«}, as illustrated opposite: SEMIGROUPS AND MONOIDS 27 4 E z„ 3áns5 3 3 2 2 2 1 1 1 0 0 0 Pi {3,4,5} Pi PA 3 4 5 For example, 26 = 2 mod 3, 26 = 2 mod 4, and 26 s 1 mod 5, so that 26<60> is represented by the section X X X Determine the congruence classes modulo 60 represented by the following sections: X (a) (b) X X X X X This page intentionally left blank I GROUPS AND QUASIGROUPS 1.

Bn. Under this multiplication, A+ becomes the so-called free semigroup over A. Adjoining an identity element 1 to A*, called the empty word, one obtains the free monoid A* over A. The length of a non-empty word a , . . an in A is the number n of "letters" appearing in it. The length of the empty word is 0. Length gives a monoid SEMIGROUPS AND MONOIDS 21 homomorphism A* -» (N, +, 0). In the notation of Exercise 4M, A" is the set of words of length n in A. 1. ,cm = dm. D This means that any word in C + can be decoded uniquely as a concatenation c, ••• c„ of codewords in C.

For which alphabets A does A* differ from A*opl IG. Can you find an example of a monoid that is not isomorphic to its opposite? IH. Let (M, •, 1) be a monoid. 7). MONOID ACTIONS 31 Fix a monoid (M, •, 1). Given two (right) M-sets (A, M), (B, M), an M-homomorphism is a function f:A-*B such that afmB = amAf for all a in A and m in A/. An isomorphism of M-sets is a bijective M-homomorphism. 4, M) and (J5, Af ) are isomorphic if there is an isomorphism / : (A, M) -» ( 5 , A/). One writes (A, M) s ( 5 , A/) in this case.