nLab lattice


This entry is about the notion in order theory/logic. For other notions of the same name, such as in bilinear form-theory, see at lattice (disambiguation).



A lattice is a poset which admits all finite meets and finite joins (or all finite products and finite coproducts, regarding a poset as a category (a (0,1)-category)).

As (0,1)-limits are meets and (0,1)-colimits are joins, a lattice is a (0,1)-category that is finitely complete and finitely cocomplete.

A lattice can also be defined as an algebraic structure, with the binary operations \wedge and \vee and the constants \top and \bot. (These correspond, respectively, to binary and nullary meets and joins in the poset-theoretic definition; accordingly, they are read ‘meet’, ‘join’, ‘top’, and ‘bottom’.) Here are the axioms for these operations:

  • \wedge and \vee are each idempotent, commutative, and associative;
  • the absorption laws: a(ab)=aa \vee (a \wedge b) = a, and a(ab)=aa \wedge (a \vee b) = a;
  • \top and \bot are the respective identities of \wedge and \vee.

You can recover the original poset from either the meet or the join; aba \leq b iff ab=aa \wedge b = a, and bab \leq a iff ab=aa \vee b = a, and then prove that aba\wedge b is the greatest lower bound for aa, bb and aba \vee b is the least upper bound for aa, bb. (Notice that the absorption laws guarantee that these two descriptions of \leq agree.) Indeed, we may say that a lattice is a bisemilattice in that it has two semilattice structures that are compatible in that they define (but in dual ways) the same partial order.

Note that a poset with only finite meets or finite joins is a (meet- or join-) semilattice, while a lattice which has all joins and meets (not just finitary ones) is a complete lattice.

Bounded lattices and pseudolattices

Traditionally, a lattice need have only finite inhabited meets and joins; that is, it need not have a top or bottom element. Algebraically, this means \wedge and \vee need not have identities.

Then one may call a lattice that does have a top and a bottom a bounded lattice; in general, a bounded poset is a poset that has top and bottom elements.

The other approach is to define a lattice, as above, to require a top and a bottom and then use the term pseudolattice to allow for the possibility that it might not.

From an algebraic point of view, requiring top and bottom is quite natural, a special case of preferring monoids to more general semigroups. In any case, one can formally adjoin a top and a bottom if required. On the other hand, many examples, especially from analysis, do not come with a top or a bottom, and adjoining them would break the other structure. For example, adjoining top (\infty) and bottom (-\infty) to the real line makes it no longer a field (addition is especially problematic); more generally, a Banach lattice need not (and, except in one degenerate case, cannot) have a top or a bottom. In fact, adjoining top and bottom to any totally ordered ring makes it no longer a ring.

Lattice homomorphisms

A lattice homomorphism ff from a lattice AA to a lattice BB is a function from AA to BB (seen as sets) that preserves \wedge and \vee (and \top and \bot, if these are required):

f(xy)=f(x)f(y),f()=,f(xy)=f(x)f(y),f()=. f(x \wedge y) = f(x) \wedge f(y),\; f(\top) = \top,\; f(x \vee y) = f(x) \vee f(y),\; f(\bot) = \bot .

Note that such a homomorphism is necessarily a monotone function, but the converse fails.

Thus, a lattice is a poset (or even a semilattice) with property-like structure.

Lattices and lattice homomorphims form a concrete category Lat.


Last revised on February 23, 2024 at 19:53:37. See the history of this page for a list of all contributions to it.