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 atlattice (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)).

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, with respective identities $\top$ and $\bot$;
- the
*absorption laws*: $a \vee (a \wedge b) = a$, and $a \wedge (a \vee b) = a$.

You can recover the original poset from either the meet or the join; $a \leq b$ iff $a \wedge b = a$, and $a \geq b$ iff $a \vee b = a$. The absorption laws guarantee that these 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.

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.

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

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

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.

Revised on January 17, 2014 11:58:51
by Urs Schreiber
(89.204.135.27)