In mathematics, a cancellative semigroup (also called a cancellation semigroup) is a semigroup having the cancellation property.[1] 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 canceled out is appearing as the left factors of a · b and a · c and hence it is a case of left cancellation property. The right cancellation property can be defined in an analogous way. Prototypical examples of cancellative semigroups are the groups and the semigroup of 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, (Suschkewitsch 1928).[2]
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Let S be a semigroup. An element a in S is left cancellative (or, is 'left cancellable', or, has the 'left cancellation property') if ab = ac implies b = c 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 ba = ca implies b = c 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.[3]
It is an elementary result in group theory that a finite cancellatve semigroup is a group. Let S be a finite cancellative semigroup. Cancellativity and finiteness taken together imply that Sa = aS = S for all a in S. So given an element a in S, there is an element ea, depending on a, in S such that aea = a. Cancellativity now further implies that this ea is independent of a and that xea = eax = x for all x in S. Thus ea is the identity element of S which may from now on be denoted by e. Using the property Sa = S one now sees that there is b in S such that ba = e. 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.
A commutative semigroup can be embedded in 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, (Clifford & Preston 1961, p. 34). 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. To obtain a sufficient 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 (Dubreil 1941) 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, (Clifford & Preston 1961, p. 35).
The first set of necessary and sufficient conditions for the embeddability of a semigroup in a group were given in (Malcev 1939).[4] Though theoretically important, the conditions are countably infinite in number and no finite subset will suffice, as shown in (Malcev 1940).[5]