special elements in a lattice

Let L be a latticeMathworldPlanetmath and aL is said to be

  • distributive if a(bc)=(ab)(ac),

  • standard if b(ac)=(ba)(bc), or

  • neutral if (ab)(bc)(ca)=(ab)(bc)(ca)

for all b,cL. There are also dual notions of the three types mentioned above, simply by exchanging and in the definitions. So a dually distributive element aL is one where a(bc)=(ab)(ac) for all b,cL, and a dually standard element is similarly defined. However, a dually neutral element is the same as a neutral element.

Remarks For any aL, suppose P is the property in L such that aP iff ab=ac and ab=ac imply b=c for all b,cL.

  • A standard element is distributive. Conversely, a distributive satisfying P is standard.

  • A neutral element is distributive (and consequently dually distributive). Conversely, a distributive and dually distributive element that satisfies P is neutral.


  • 1 G. Birkhoff Lattice Theory, 3rd Edition, AMS Volume XXV, (1967).
  • 2 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998).
Title special elements in a lattice
Canonical name SpecialElementsInALattice
Date of creation 2013-03-22 16:42:29
Last modified on 2013-03-22 16:42:29
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Definition
Classification msc 06B99
Defines distributive element
Defines standard element
Defines neutral element
Defines dually distributive
Defines dually standard