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Let be a semigroup.
1 Regular semigroups
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 is regular, so that for some . Put . Then
so is an inverse of .
2 Inverse semigroups
is an inverse semigroup if for all there is a unique such that and .
In an inverse semigroup every principal ideal is generated by a unique idempotent.
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.
is called eventually regular (or -regular) if a power of every element is regular.
is called group-bound (or strongly -regular, or an epigroup) if a power of every element is in a subgroup of .
is called completely regular if every element is in a subgroup of .
|Date of creation||2013-03-22 14:23:17|
|Last modified on||2013-03-22 14:23:17|
|Last modified by||yark (2760)|