nLab
semilattice

Contents

Definition

A meet-semilattice is a poset which admits all finite meets, or equivalently which admits a top element \top and binary meets aba\wedge b.

In a meet-semilattice the binary meet \wedge is commutative, associative, has \top as a unit, and is idempotent: aa=aa\wedge a =a. And in fact, given any commutative and idempotent monoid (A,,)(A,\wedge,\top), we can define aba\le b to mean ab=aa \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.

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.

Bounded semilattices and semipseudolattices

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.

Semilattice homomorphisms

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

f(xy)=f(x)f(y),f()=. 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.

Examples

Revised on March 15, 2012 17:10:42 by Urs Schreiber (82.169.65.155)