# 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 $\varnothing$, 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 $|A|\leq\kappa$, $\bigvee A$ exists. If $\kappa$ is finite, then $L$ is just a join-semilattice. When $\kappa=\infty$, 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 $a\leq b$ in $A$, then $a\leq b$ in $L$

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

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

• if $a\in A$, $c\in B$, then $c\leq a$ 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 $a\neq 0$ 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 $f(a)\wedge f(b)=1\wedge 1=1$.

## References

• 1 G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott, Continuous Lattices and Domains, Cambridge University Press, Cambridge (2003).
• 2 P. T. Johnstone, Stone Spaces, Cambridge University Press (1982).
 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