By Professor Richard Hubert Bruck (auth.)

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**Example text**

Call 5 quasi-regular on the left if, whenever two elements a, b of 5 have a common left multiple, there exist elements c, d in 5 such that ca = db and at least one of c, d is regular on the left. Doss shows that a cancellation semigroup which is quasi-regular on the left can be imbedded in a group. This result contains as a special case an earlier one of DuBREIL [186] which uses the Öre condition: every element of 5 is regular on the right. Additional literatme: PTAK [244], [245], DUBREIL-jACOTIN [190], KüNTOROVIC [209].

Therefore G0 is also the set of all elements x-1 y where x, y range over G. Since (x-1 y)- 1 = y- 1 x, we conclude that G0 is a subgroup of K. If x is in G and p is in G0 , then p = y 2 •••• Yn is a product of n- 1 elements of G; thus x-1 px = (x-1 y 2) (y3 ••• YnX) is the product of two elements of G0 and hence is in G0 • That is, x-1 G0 x C G0 • Similarly, we may show first that G0 contains all elements y x-1 with x, y in G and thence that xG0 x-1 C G0 for every x in G. Now it clear from (i) that k-1 G0 k = G0 for every k in K.

Hence () = () cx ß, ()' = ()' ß cx. lf x is in H, x = afJ for at least one a in G. Then xcxß = afJcxß = a() = x, so cx ß = I H· Similarly, ß cx = I H'· Consequently cx and ß are one-to-one upon. Therefore cx is an isomorphism of H upon H'. (II) Existence. We give two conceptually different proofs for the case of homomorphisms upon cancellation groupoids. Similar proofs 4. Hornamorphie imbedding 33 hold for homomorphisms upon semigroups or upon cancellation semigroups. The first proof requires us to accept the existence of the dass (/> of all homomorphisms of G upon cancellation groupoids.