one-sided normality of subsemigroup

Let S be a semigroupPlanetmathPlanetmath. A subsemigroup N of S is said to be left-normal if gNNg for all gS and it is said to be right-normal if gNNg for all gS. One may similarly define left-normalizers


and right-normalizers


A left-normal subgroupMathworldPlanetmath N of a group S is automatically normal, since


In is similarly shown for general S and N that if some gLNS(N) has an inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath g-1 then g-1RNS(N) and vice versa. Left- and right-normalizers are always closed underPlanetmathPlanetmath multiplication (hence subsemigroups) and contain the identity elementMathworldPlanetmath of S if there is one.

An example of a left-normal but not right-normal NS can be constructed using matrices under multiplication, if one takes

S={(km01)|k,m}  and  N={(1n01)|n},

where one may note that N is a group and S is a monoid. Since

(km01)(1n01)= (kkn+m01)and
(1n01)(km01)= (kn+m01)

it follows that gNNg for all gS, with proper inclusion ( when k±1.

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 combinationMathworldPlanetmathPlanetmath ‘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
Canonical name OnesidedNormalityOfSubsemigroup
Date of creation 2013-03-22 16:10:41
Last modified on 2013-03-22 16:10:41
Owner lars_h (9802)
Last modified by lars_h (9802)
Numerical id 6
Author lars_h (9802)
Entry type Definition
Classification msc 20A05
Defines left-normal
Defines right-normal
Defines left-normalizer
Defines right-normalizer