A latticoid is a set L with two binary operationsMathworldPlanetmath, the meet and the join on L satisfying the following conditions:

  1. 1.

    (idempotence) xx=xx=x for any xL,

  2. 2.

    (commutativity) xy=yx and xy=yx for any x,yL, and

  3. 3.

    (absorption) x(yx)=x(yx)=x for any x,yL.

A latticoid is like a lattice without the associativity assumptionPlanetmathPlanetmath, i.e., a lattice is a latticoid that is both meet associative and join associative.

If one of the binary operations is associative, say is associative, we may define a latticoid as a poset as follows:

xy iff xy=x.

Clearly, is reflexiveMathworldPlanetmathPlanetmath, as xx=x. If xy and yx, then x=xy=yx=y, so is anti-symmetric. Finally, suppose xy and yz, then xz=(xy)z=x(yz)=xy=x, or xz, is transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath.

Once a latticoid is a poset, we may easily visualize it by a diagram (Hasse diagram), much like that of a lattice. Position y above x if xy and connect a line segment between x and y. The following is the diagram of a latticoid that is meet associative but not join associative:


It is not join associative because (ab)c=ab, whereas a(bc)=ac=cab.

Given a latticoid L, we can define a dual L* of L by using the same underlying set, and define the meet of a and b in L* as the join of a and b in L, and the join of a and b (in L*) as the meet of a and b in L. L is a meet-associative latticoid iff L* is join-associative.

Title latticoid
Canonical name Latticoid
Date of creation 2013-03-22 16:31:02
Last modified on 2013-03-22 16:31:02
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 06F99