lattice ideal

Let L be a latticeMathworldPlanetmath. An ideal I of L is a non-empty subset of L such that 

  1. 1.

    I is a sublattice of L, and

  2. 2.

    for any aI and bL, abI.

Note the similarity between this definition and the definition of an ideal ( in a ring (except in a ring with 1, an ideal is almost never a subring)

Since the fact that abI for a,bI in the first condition is already implied by the second condition, we can replace the first condition by a weaker one:

for any a,bI, abI.

Another equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath characterization of an ideal I in a lattice L is

  1. 1.

    for any a,bI, abI, and

  2. 2.

    for any aI, if ba, then bI.

Here’s a quick proof. In fact, all we need to show is that the two second conditions are equivalent for I. First assume that for any aI and bL, abI. If ba, then b=abI. Conversely, since abaI, abI as well.

Special Ideals. Let I be an ideal of a lattice L. Below are some common types of ideals found in lattice theory.

  • I is proper if IL.

  • If L contains 0, I is said to be non-trivial if I0.

  • I is a prime idealMathworldPlanetmathPlanetmathPlanetmath if it is proper, and for any abI, either aI or bI.

  • I is a maximal idealMathworldPlanetmath of L if I is proper and the only ideal having I as a proper subsetMathworldPlanetmathPlanetmath is L.

  • ideal generated by a set. Let X be a subset of a lattice L. Let S be the set of all ideals of L containing X. Since S (LS), the intersectionMathworldPlanetmathPlanetmath M of all elements in S, is also an ideal of L that contains X. M is called the ideal generated by X, written (X]. If X is a singleton {x}, then M is said to be a principal idealMathworldPlanetmathPlanetmathPlanetmath generated by x, written (x]. (Note that this construction can be easily carried over to the construction of a sublattice generated by a subset of a lattice).

Remarks. Let L be a lattice.

  1. 1.

    Given any subset XL, let X be the set consisting of all finite joins of elements of X, which is clearly a directed setMathworldPlanetmath. Then X, the down set of X, is (X]. Any element of (X] is less than or equal to a finite join of elements of X.

  2. 2.

    If L is a distributive latticeMathworldPlanetmath, every maximal ideal is prime. Suppose IL is maximal and abI with aI. Then the ideal generated by I and a must be L, so that bpa for some pI. Then b=bb(pa)b=(pb)(ab)I, which means bI. So I is prime.

  3. 3.

    If L is a complemented latticeMathworldPlanetmath, every prime ideal is maximal. Suppose IL is prime and aI. Let b be a complement of a, then bI, for otherwise, 0=abI, a contradictionMathworldPlanetmathPlanetmath. Let J be the ideal generated by I and a, then 1baJ, so J=L.

  4. 4.

    Combining the two results above, in a Boolean algebraMathworldPlanetmath, an ideal is prime iff it is maximal.

Examples. In the lattice L below,


Besides L and {0}, below are all proper idealsMathworldPlanetmath of L:

  • M={a,c,d,e,f,0},

  • N={b,c,d,e,f,0},

  • R={c,d,e,f,0},

  • S={d,e,f,0},

  • T={e,0}, and

  • U={f,0}.

Out of these, M,N,S,T,U are prime, and M,N are maximal. The ideal generated by, say {c,e}, is R. Looking more closely, we see that R can actually be generated by c, and so is principal. In fact, all ideals in L are principal, generated by their maximal elementsMathworldPlanetmath. It is not hard to see, that in a lattice L with acc (ascending chain conditionMathworldPlanetmathPlanetmathPlanetmath), all ideals are principal:


. First, let’s show that an ideal I in a lattice L with acc has at least one maximal element. Suppose aI. If a is not maximal in I, there is a a1I such that aa1. If a1 is not maximal in I, repeat the process above so we get a chain aa1a2 in I. Eventually this chain terminates an=an+1=. Thus b=an is maximal in I. Next, suppose that I has two distinct maximal elements. Then their join is again in I, contradicting maximality. So b is unique and all elements c such that cb must be in I. Therefore, I=(b].∎

Finally, an example of a sublattice that is not an ideal is the subset {b,c,d,e,0}.

Title lattice ideal
Canonical name LatticeIdeal
Date of creation 2013-03-22 15:48:58
Last modified on 2013-03-22 15:48:58
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 17
Author CWoo (3771)
Entry type Definition
Classification msc 06B10
Synonym prime lattice ideal
Synonym maximal lattice ideal
Related topic LatticeFilter
Related topic UpperSet
Related topic OrderIdeal
Related topic LatticeOfIdeals
Defines ideal
Defines proper ideal
Defines prime ideal
Defines sublattice generated by
Defines ideal generated by
Defines principal ideal
Defines maximal ideal
Defines non-trivial ideal