nLab
lattice

Definition

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 lattice can also be defined as an algebraic structure, with the binary operations $\wedge$ and $\vee$ and the constants $\top$ and $\perp$. (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 $\perp$;
• 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; $a\le b$ iff $a\wedge b=a$, and $a\ge b$ 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.

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 ($\mathrm{\infty }$) and bottom ($-\mathrm{\infty }$) 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.

Lattice homomorphisms

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 $\perp$, if these are required):

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