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).

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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 (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: $a \vee (a \wedge b) = a$, and $a \wedge (a \vee b) = a$;
• $\top$ and $\bot$ are the respective identities of $\wedge$ and $\vee$.

You can recover the original poset from either the meet or the join; $a \leq b$ iff $a \wedge b = a$, and $b \leq a$ iff $a \vee b = a$, and then prove that $a\wedge b$ is the greatest lower bound for $a$, $b$ and $a \vee b$ is the least upper bound for $a$, $b$. (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 $f$ 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.

Related concepts

• $\sigma$-complete lattice

• complete lattice

• distributive lattice

• modular lattice

• orthomodular lattice

• geometric lattice?

• continuous lattice

• complemented lattice

• semilattice

• suplattice

• Hilbert lattice

• frame

• lattice-ordered abelian group

• lattice object

References

• Peter Johnstone, chapter 1 of Stone Spaces

• Jacob Lurie, section A.1.1 of Spectral Algebraic Geometry