By P.A. Grillet

This is often the 1st booklet approximately commutative semigroups normally. Emphasis is on constitution however the different elements of the idea are a minimum of surveyed and a whole set of approximately 850 references is integrated. The booklet is meant for mathematicians who do study on semigroups or who come upon commutative semigroups of their examine.

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**Extra info for Commutative Semigroups**

**Example text**

P. p. 6 show that all ideal extensions of S can be constructed in this fashion. 7. s. S by a semigroup Q with zero is determined by a partial homomorphism of Q\ {0} into G(S), namely the restriction to Q\0 ofthe canonical homomorphism a f----7 axjx = xajx, where x is any element of S. Stronger results were obtained by Heuer & Miller [ 1966] and Heuer [ 1971] in case Q is a group with a zero element adjoined. 3. s. have been studied in various ways. s. S has a (topological) uniform structure. Schein [ 197 5] showed that the divisibility preorder ;£J{ on S induces a compatible preorder ;£ on G(S) and showed that every subsemigroup of S is densely embedded in its idealizer if and only if G (S) is archimedean under this preorder.

Proof. Subsemigroups of groups are cancellative. 2, and S can be embedded into the abelian group G(S). 0 This provides the familiar embedding of S into an abelian group, namely, the group of fractions or universal group G(S) of S. When S is cancellative it is standard practice to identify x E S and o:(x) E G(S); then xja becomes xa- 1 and S becomes a subsemigroup of G(S). s. enjoy additional properties. 3. s. An abelian group G is isomorphic to the universal group of S if and only if S is isomorphic to a subsemigroup T of G such that every g E G can be written in the form g = ab- 1 for some a,b E T.

UNIVERSAL GROUPS. for some y,t E S. Hence axyt surjective. = xt and xla = xytlt = 1(xy). Thus 1 is If conversely 1 is surjective and x E S, then 1( x) - 1 yES, = xylx, y and t = x 2 u E S. 5 below implies that G(S) is isomorphic to the kernel of S. If on the other hand S = N, or more generally if S is cancellative but not a group, then 1 is not surjective, since it is injective but not bijective. 4. For future reference we note the following properties. 5. When I is a nonempty ideal of a commutative semigroup S, then the inclusion homomorphism I--+ S induces an isomorphism G(I) ~ G(S).