Cancellative semigroup

In mathematics, a cancellative semigroup (also called a cancellation semigroup) is a semigroup having the cancellation property. In intuitive terms, the cancellation property asserts that from an equality of the form "a"·"b" = "a"·"c", where · is a binary operation, one can cancel the element "a" and deduce the equality "b" = "c". In this case the element being cancelled out is appearing as the left factors of and and hence it is a case of the left cancellation property. The right cancellation property can be defined analogously. Prototypical examples of cancellative semigroups are the positive integers under addition or multiplication. Cancellative semigroups are considered to be very close to being groups because cancellability is one of the necessary conditions for a semigroup to be embeddable in a group. Moreover, every finite cancellative semigroup is a group. One of the main problems associated with the study of cancellative semigroups is to determine the necessary and sufficient conditions for embedding a cancellative semigroup in a group.
The origins of the study of cancellative semigroups can be traced to the first substantial paper on semigroups, .
Formal definitions.
Let "S" be a semigroup. An element "a" in "S" is left cancellative (or, is "left cancellable", or, has the "left cancellation property") if implies for all "b" and "c" in "S". If every element in "S" is left cancellative, then "S" is called a left cancellative semigroup.
Let "S" be a semigroup. An element "a" in "S" is right cancellative (or, is "right cancellable", or, has the "right cancellation property") if implies for all "b" and "c" in "S". If every element in "S" is right cancellative, then "S" is called a right cancellative semigroup.
Let "S" be a semigroup. If every element in "S" is both left cancellative and right cancellative, then "S" is called a cancellative semigroup.
Alternative definitions.
It is possible to restate the characteristic property of a cancellative element in terms of a property held by the corresponding left multiplication and right multiplication maps defined by and . An element "a" in "S" is left cancellative if and only if "L""a" is injective. An element "a" is right cancellative if and only if "R""a" is injective.
Finite cancellative semigroups.
It is an elementary result in group theory that a finite cancellative semigroup is a group. Let "S" be a finite cancellative semigroup. Cancellativity and finiteness taken together imply that for all "a" in "S". So given an element "a" in "S", there is an element "e""a", depending on "a", in "S" such that . Cancellativity now further implies that this "e""a" is independent of "a" and that for all "x" in "S". Thus "e""a" is the identity element of "S", which may from now on be denoted by "e". Using the property one now sees that there is "b" in "S" such that . Cancellativity can be invoked to show that "ab" = "e" also, thereby establishing that every element "a" in "S" has an inverse in "S". Thus "S" must necessarily be a group.
Furthermore, every cancellative epigroup is also a group.
Embeddability in groups.
A commutative semigroup can be embedded in a group (i.e., is isomorphic to a subset of a group) if and only if it is cancellative. The procedure for doing this is similar to that of embedding an integral domain in a field, . See also Grothendieck group, the universal mapping from a commutative semigroup to abelian groups that is an embedding if the semigroup is cancellative.
For the embeddability of noncommutative semigroups in groups, cancellativity is obviously a necessary condition. However, it is not sufficient: there are (noncommutative and infinite) cancellative semigroups that cannot be embedded in a group.
To obtain a sufficient (but not necessary) condition, it may be observed that the proof of the result that a finite cancellative semigroup "S" is a group critically depended on the fact that "Sa" = "S" for all "a" in "S". The paper generalized this idea and introduced the concept of a "right reversible" semigroup. A semigroup "S" is said to be "right reversible" if any two principal ideals of "S" intersect, that is, "Sa" ∩ "Sb" ≠ Ø for all "a" and "b" in "S". The sufficient condition for the embeddability of semigroups in groups can now be stated as follows: (Ore's Theorem) Any right reversible cancellative semigroup can be embedded in a group, .
The first set of necessary and sufficient conditions for the embeddability of a semigroup in a group were given in . Though theoretically important, the conditions are countably infinite in number and no finite subset will suffice, as shown in . A different (but also countably infinite) set of necessary and sufficient conditions were given in , where it was shown that a semigroup can be embedded in a group if and only if it is cancellative and satisfies a so-called "polyhedral condition". The two embedding theorems by Malcev and Lambek were later compared in .