complete semilattice

A complete join-semilattice is a join-semilattice $L$ such that for any subset $A\subseteq L$, $\bigvee A$, the arbitrary join operation on $A$, exists. Dually, a complete meet-semilattice is a meet-semilattice such that $\bigwedge A$ exists for any $A\subseteq L$. Because there are no restrictions placed on the subset $A$, 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 $\bigvee$ ($\bigwedge$) applies to any set, including $L$ itself, and the empty set $\mathrm{\varnothing }$, so that $L$ 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 $A\subseteq L$ to be countable, we get what is a called a countably complete join-semilattice (or dually, a countably complete meet-semilattice). More generally, if $\kappa$ is any cardinal, then a $\kappa$-complete join-semilattice is a semilattice $L$ such that for any set $A\subseteq L$ such that $|A|\le \kappa$, $\bigvee A$ exists. If $\kappa$ is finite, then $L$ is just a join-semilattice. When $\kappa =\mathrm{\infty }$, the only requirement on $A\subseteq L$ is that it be non-empty. In [1], a complete semilattice is defined to be a poset $L$ such that for any non-empty $A\subseteq L$, $\bigwedge A$ exists, and any directed set $D\subseteq L$, $\bigvee D$ exists.

Example. Let $A$ and $B$ be two isomorphic complete chains (a chain that is a complete lattice) whose cardinality is $\kappa$. Combine the two chains to form a lattice $L$ by joining the top of $A$ with the top of $B$, and the bottom of $A$ with the bottom of $B$, so that

• if $a\le b$ in $A$, then $a\le b$ in $L$

• if $c\le d$ in $B$, then $c\le d$ in $L$

• if $a\in A$, $c\in B$, then $a\le c$ iff $a$ is the bottom of $A$ and $c$ is the top of $B$

• if $a\in A$, $c\in B$, then $c\le a$ iff $a$ is the top of $A$ and $c$ is the bottom of $B$

Now, $L$ can be easily seen to be a $\kappa$-complete lattice. Next, remove the bottom element of $L$ to obtain $L^{\prime }$. Since, the meet operation no longer works on all pairs of elements of $L^{\prime }$ while $\vee$ still works, $L^{\prime }$ is a join-semilattice that is not a lattice. In fact, $\bigvee$ works on all subsets of $L^{\prime }$. Since $|L^{\prime }|=\kappa$, we see that $L^{\prime }$ is a $\kappa$-complete join-semilattice.

Remark. Although a complete semilattice is the same as a complete lattice, a homomorphism $f$ between, say, two complete join-semilattices $L_1$ and $L_2$, may fail to be a homomorphism between $L_1$ and $L_2$ as complete lattices. Formally, a complete join-semilattice homomorphism between two complete join-semilattices $L_1$ and $L_2$ is a function $f:L_1\to L_2$ such that for any subset $A\subseteq L_1$, we have

$$f(\bigvee A)=\bigvee f(A)$$

where $f(A)=\{f(a)\mid a\in A\}$. Note that it is not required that $f(\bigwedge A)=\bigwedge f(A)$, so that $f$ needs not be a complete lattice homomorphism.

To give a concrete example where a complete join-semilattice homomorphism $f$ fails to be complete lattice homomorphism, take $L$ from the example above, and define $f:L\to L$ by $f(a)=1$ if $a\ne 0$ and $f(0)=0$. Then for any $A\subseteq L$, it is evident that $f(\bigvee A)=\bigvee f(A)$. However, if we take two incomparable elements $a,b\in L$, then $f(a\wedge b)=f(0)=0$, while $f(a)\wedge f(b)=1\wedge 1=1$.

Title	complete semilattice
Canonical name	CompleteSemilattice
Date of creation	2013-03-22 17:44:49
Last modified on	2013-03-22 17:44:49
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	13
Author	CWoo (3771)
Entry type	Definition
Classification	msc 06A12
Classification	msc 06B23
Synonym	countably complete upper-semilattice
Synonym	countably complete lower-semilattice
Synonym	complete upper-semilattice homomorphism
Synonym	complete lower-semilattice homomorphism
Related topic	CompleteLattice
Related topic	Semilattice
Related topic	ArbitraryJoin
Defines	countably complete join-semilattice
Defines	countably complete meet-semilattice
Defines	complete join-semilattice homomorphism
Defines	complete meet-semilattice homomorphism