one-sided normality of subsemigroup
A left-normal subgroup of a group is automatically normal, since
In is similarly shown for general and that if some has an inverse then and vice versa. Left- and right-normalizers are always closed under multiplication (hence subsemigroups) and contain the identity element of if there is one.
An example of a left-normal but not right-normal can be constructed using matrices under multiplication, if one takes
where one may note that is a group and is a monoid. Since
it follows that for all , with proper inclusion (http://planetmath.org/ProperSubset) when .
The definition of left and is somewhat arbitrary in the choice of whether to call something the or left form. A reference supporting the choice documented here is:
- 1 Karl Heinrich Hofmann and Michael Mislove: The centralizing theorem for left normal groups of units in compact monoids, Semigroup Forum 3 (1971/72), no. 1, 31–42.
It may also be observed that the combination ‘left normal’ in semigroup theory frequently occurs as part of the phrase ‘left normal band’, but in that case the etymology rather seems to be that ‘left’ qualifies the phrase ‘normal band’.
|Title||one-sided normality of subsemigroup|
|Date of creation||2013-03-22 16:10:41|
|Last modified on||2013-03-22 16:10:41|
|Last modified by||lars_h (9802)|