By T.Y. Lam

By way of aiming the extent of writing on the amateur instead of the gourmand and via stressing the function of examples and motivation, the writer has produced a textual content that's compatible for a one-semester graduate direction or for self-study.

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24 5 2. Ch. 1. e. with an operation a : X x X -+ X such that (A) a(cr(x,y), z ) = a(x, a(y, 2 ) ) for any x, y , z in X . g. as x . , or simply by juxtaposition as xy. The associativity condition (A) then takes the usual form (xy) z = x(yz). A monoid is a semigroup with a nullary operation e, called unit, such that for every x, xe = ex = x. 2. ). This will, however, be done if really necessary. If there is no danger of confusion, we will usually use the same symbol for the semigroup (monoid) and for its underlying set (the X above).

7 allows us to go a step further. Define right translations R,: M -+ M putting R,(x) = xa. Consider the R,‘s as unary operations on M . Take notice of the following facts: I. 5). Indeed, we have Rb(L,(x))= (ax)b = a(xb) = L,(Rb(x)). 11. , for every a, R, o f = f o R,. Hence, we have f ( x ) = f(R,(e)) = = R,(f(e)) = f ( e )x. Combining I and I1 we obtain --f Theorem. For every monoid M there is a set X and a collection (c&, of unary operations such that the monoid of all mappings which are simultaneously ciicci-homomorphismsfor all i is isomorphic to M .

6. Taking an element a =I= A and putting B = A u (a}, we can represent A-ary operations as B-ary relations putting, for a : X A + X , z = {< E xB1 t ( a ) = a(< 1 A ) ) . This representation preserves the choice of homomorphisms: Really, if f: X Y is an ag-homomorphism and < E Z , we have -+ (f 4 (4 = f ( W )= f ( 4 4 1 A ) ) = P ( f 0 (i" I 4)= P((f0 4;) I 4 < a. 7 so that f o E If f is an $-homomorphism and 5: A + X , define q : B -+ X by putting q A = 4 and ~ ( a=) a(<). 7. The identity mapping is always an rr-homomorphism.

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