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Download PDF by Thomas Keilen: Algebra I. Lecture Notes

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This map is called the canonical monoid structure on 9 since it is the unique monoid structure on P which makes the canonical surjective map k»:X--»9 a morphism of monoids. 40 TWO/MONOIDS AND GROUPS If 9 is a partition of a monoid X, then we shall denote the monoid (9, m) consisting of the set 9 together with the canonical monoid structure m simply by 9. It should be noted that if 9 is a partition of the monoid X, then the canonical monoid structure on 9 is completely described by the appealing formula [x,][x2] = [x,x2] for all x, and x2 in X.

The appropriateness of this remark is reinforced by the following. 1 The following conditions are equivalent for a partition 9 of the underlying set of a monoid X: (a) If Xl and X2 are elements of 9, then there is one (and consequently only one) element X3 in 9 containing X,X2. (b) There exists one (and consequently only one) map m: 9 x 9 -» 9 such that the canonical surjective map k&:X-»9 has the property kg(x\x2) = m(fc»(X1). Mjfc)) for all x, and x2 in X. :X -» 9 is a morphism of monoids.

Suppose f:X-»Y is a morphism of monoids. org/access_use#cc-zero underlying sets of X and Y, we know that there is associated with the map / the partition Coim fofX whose elements are the subsets of X of the form /"'(y) for all y in Im /. Suppose f"\y,) and /"'(y2) are two elements of Coim /. The fact that / is a morphism of monoids implies that if x, is in f\y,) and x2 is in /_,(y2), then x,x2 is in /"'(y,y2). This is equivalent to saying that if we denote by /"'(y,)/"'(y2) the set of all elements in X of the form X,X2 with x, and x2 in / '(y2), then /"'(y,)/"'(y2)C 38 TWO/MONOIDS AND GROUPS f Xy,yi)- This condition can be restated as follows: If the subsets X, and X2 of X are elements of Coim /, then there is one and only one element X3 in Coim / con- taining X,X2 where X,X2 is the set of all elements of X of the form x,x2 with X, in X, and x2 in X2.

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Algebra I. Lecture Notes by Thomas Keilen


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