A lattice can also be defined as an algebraic structure, with the binary operations and and the constants and . (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:
You can recover the original poset from either the meet or the join; iff , and iff . 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.
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 and 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 () and bottom () 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 from a lattice to a lattice is a function from to (seen as sets) that preserves and (and and , 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.