regular semigroup (original) (raw)

x∈S is regular if there is a y∈S such that x=x⁢y⁢x.
y∈S is an inverse(or a relative inverse) for x if x=x⁢y⁢x and y=y⁢x⁢y.

1 Regular semigroups

S is a regular semigroup if all its elements are regular. The phrase ’von Neumann regular’ is sometimes used, after the definition for rings.

Every regular element has at least one inverse. To show this, suppose a∈S is regular, so that a=a⁢b⁢a for some b∈S. Put c=b⁢a⁢b. Then

and

so c is an inverse of a.

2 Inverse semigroups

S is an inverse semigroup if for all x∈S there is a unique y∈S such that x=x⁢y⁢x and y=y⁢x⁢y.

In an inverse semigroup every principal ideal is generated by a unique idempotent.

In an inverse semigroup the set of idempotents is a subsemigroup, in particular a commutative band (http://planetmath.org/ASemilatticeIsACommutativeBand).

The bicyclic semigroup is an example of an inverse semigroup. The symmetric inverse semigroup (on some set X) is another example. Of course, every group is also an inverse semigroup.

3 Motivation

Both of these notions generalise the definition of a group. In particular, a regular semigroup with one idempotent is a group: as such, many interesting subclasses of regular semigroups arise from putting conditions on the idempotents. Apart from inverse semigroups, there are orthodox semigroups where the set of idempotents is a subsemigroup, and Clifford semigroups where the idempotents are central.

4 Additional

S is called eventually regular (or π-regular) if a power of every element is regular.

S is called group-bound (or strongly π-regular, or an epigroup) if a power of every element is in a subgroup of S.

S is called completely regular if every element is in a subgroup of S.