semilattice

A **meet-semilattice** is a poset which admits all finite meets, or equivalently which admits a top element $\top$ and binary meets $a\wedge b$. If we think of a poset as a category, a meet-semilattice is the same as a poset with finite limits, or equivalently, a poset with finite products.

In a meet-semilattice the binary meet $\wedge$ is commutative, associative, has $\top$ as a unit, and is *idempotent*: $a\wedge a =a$. And in fact, given any commutative and idempotent monoid $(A,\wedge,\top)$, we can define $a\le b$ to mean $a \wedge b = a$ to make it into a poset with finite meets; thus we have an equivalent algebraic definition of a meet-semilattice.

Dually, a **join-semilattice** is a poset which admits all finite joins, including a bottom element $\bot$ and binary joins $\vee$. Once again $\vee$ is commutative, associative, unital for $\bot$, and idempotent, and we can recover the order from it. If we think of a poset as a category, a join-semilattice is the same as a poset with finite colimits, or equivalently, a poset with finite coproducts.

Note that the algebraic definition of both types of semilattice is the same: a commutative idempotent monoid. The difference comes in how we define the order, or if we work purely algebraically, in the notation we use (just as we distinguish additive and multiplicative groups notationally). It would also be possible to take one as standard and call the other a *cosemilattice* (compare directed and codirected sets), but this may not have ever been done.

If a poset is both a meet- *and* a join-semilattice, then we call it a lattice.

Traditionally, a semilattice need have only finite inhabited meets/joins; that is, it need not have a top/bottom element. Algebraically, this means that a semilattice need not be a monoid, but is any commutative idempotent semigroup.

One might call a semilattice that does have a top/bottom element a **bounded semilattice**; the problem with this is that a bounded poset already means a poset that has both top *and* bottom elements, whereas here we really only want to require one.

Another approach is to define a semilattice, as above, to require a top/bottom element and then use the term **pseudosemilattice** or **semipseudolattice** to allow for the possibility that it might not.

See lattice for more discussion of this issue.

A semilattice homomorphism $f$ from a semilattice $A$ to a semilattice $B$ is a function from $A$ to $B$ (seen as sets) that preserves $\vee$ (and $\bot$, if this is required):

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

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

Thus, a semilattice is a poset with property-like structure.

Semilattices and semilattice homomorphims form a concrete category SemiLat.

Revised on August 23, 2017 01:32:09
by John Baez
(137.132.224.134)