A complete join-semilattice is a join-semilattice such that for any subset , , the arbitrary join operation on , exists. Dually, a complete meet-semilattice is a meet-semilattice such that exists for any . Because there are no restrictions placed on the subset , it turns out that a complete join-semilattice is a complete meet-semilattice, and therefore a complete lattice. In other words, by dropping the arbitrary join (meet) operation from a complete lattice, we end up with nothing new. For a proof of this, see here (http://planetmath.org/CriteriaForAPosetToBeACompleteLattice). The crux of the matter lies in the fact that () applies to any set, including itself, and the empty set , so that always contains has a top and a bottom.
Variations. To obtain new objects, one looks for variations in the definition of “complete”. For example, if we require that any to be countable, we get what is a called a countably complete join-semilattice (or dually, a countably complete meet-semilattice). More generally, if is any cardinal, then a -complete join-semilattice is a semilattice such that for any set such that , exists. If is finite, then is just a join-semilattice. When , the only requirement on is that it be non-empty. In , a complete semilattice is defined to be a poset such that for any non-empty , exists, and any directed set , exists.
Example. Let and be two isomorphic complete chains (a chain that is a complete lattice) whose cardinality is . Combine the two chains to form a lattice by joining the top of with the top of , and the bottom of with the bottom of , so that
if in , then in
if in , then in
if , , then iff is the bottom of and is the top of
if , , then iff is the top of and is the bottom of
Now, can be easily seen to be a -complete lattice. Next, remove the bottom element of to obtain . Since, the meet operation no longer works on all pairs of elements of while still works, is a join-semilattice that is not a lattice. In fact, works on all subsets of . Since , we see that is a -complete join-semilattice.
Remark. Although a complete semilattice is the same as a complete lattice, a homomorphism between, say, two complete join-semilattices and , may fail to be a homomorphism between and as complete lattices. Formally, a complete join-semilattice homomorphism between two complete join-semilattices and is a function such that for any subset , we have
where . Note that it is not required that , so that needs not be a complete lattice homomorphism.
To give a concrete example where a complete join-semilattice homomorphism fails to be complete lattice homomorphism, take from the example above, and define by if and . Then for any , it is evident that . However, if we take two incomparable elements , then , while .
|Date of creation||2013-03-22 17:44:49|
|Last modified on||2013-03-22 17:44:49|
|Last modified by||CWoo (3771)|
|Synonym||countably complete upper-semilattice|
|Synonym||countably complete lower-semilattice|
|Synonym||complete upper-semilattice homomorphism|
|Synonym||complete lower-semilattice homomorphism|
|Defines||countably complete join-semilattice|
|Defines||countably complete meet-semilattice|
|Defines||complete join-semilattice homomorphism|
|Defines||complete meet-semilattice homomorphism|