algebraic definition of a lattice

The parent entry ( defines a lattice as a relational structure (a poset) satisfying the condition that every pair of elements has a supremumMathworldPlanetmathPlanetmath and an infimumMathworldPlanetmath. Alternatively and equivalently, a lattice L can be a defined directly as an algebraic structurePlanetmathPlanetmath with two binary operationsMathworldPlanetmath called meet and join satisfying the following conditions:

  • (idempotency of and ): for each aL, aa=aa=a;

  • (commutativity of and ): for every a,bL, ab=ba and ab=ba;

  • (associativity of and ): for every a,b,cL, a(bc)=(ab)c and a(bc)=(ab)c; and

  • (absorption): for every a,bL, a(ab)=a and a(ab)=a.

It is easy to see that this definition is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the one given in the parent, as follows: define a binary relationMathworldPlanetmath on L such that

ab iff ab=b.

Then is reflexiveMathworldPlanetmathPlanetmath by the idempotency of . Next, if ab and ba, then a=ab=b, so is anti-symmetric. Finally, if ab and bc, then ac=a(bc)=(ab)c=bc=c, and therefore ac. So is transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath. This shows that is a partial orderMathworldPlanetmath on L. For any a,bL, a(ab)=(aa)b=ab so that aab. Similarly, bab. If ac and bc, then (ab)c=a(bc)=ac=c. This shows that ab is the supremum of a and b. Similarly, ab is the infimum of a and b.

Conversely, if (L,) is defined as in the parent entry, then by defining

ab=sup{a,b} and ab=inf{a,b},

the four conditions above are satisfied. For example, let us show one of the absorption laws: a(ab)=a. Let c=inf{a,b}a=ab. Then ca so that sup{a,c}=a, which precisely translates to a=ac=a(ab). The remainder of the proof is left for the reader to try.

Title algebraic definition of a lattice
Canonical name AlgebraicDefinitionOfALattice
Date of creation 2013-03-22 17:39:29
Last modified on 2013-03-22 17:39:29
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 12
Author CWoo (3771)
Entry type Definition
Classification msc 03G10
Classification msc 06B99