complete semilattice
A complete^{} joinsemilattice is a joinsemilattice $L$ such that for any subset $A\subseteq L$, $\bigvee A$, the arbitrary join operation^{} on $A$, exists. Dually, a complete meetsemilattice is a meetsemilattice 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 joinsemilattice is a complete meetsemilattice, 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 joinsemilattice (or dually, a countably complete meetsemilattice). More generally, if $\kappa $ is any cardinal, then a $\kappa $complete joinsemilattice 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 joinsemilattice. When $\kappa =\mathrm{\infty}$, the only requirement on $A\subseteq L$ is that it be nonempty. In [1], a complete semilattice is defined to be a poset $L$ such that for any nonempty $A\subseteq L$, $\bigwedge A$ 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

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

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

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

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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 joinsemilattice 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 joinsemilattice.
Remark. Although a complete semilattice is the same as a complete lattice, a homomorphism^{} $f$ between, say, two complete joinsemilattices ${L}_{1}$ and ${L}_{2}$, may fail to be a homomorphism between ${L}_{1}$ and ${L}_{2}$ as complete lattices. Formally, a complete joinsemilattice homomorphism between two complete joinsemilattices ${L}_{1}$ 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 joinsemilattice 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 $f(a)\wedge f(b)=1\wedge 1=1$.
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  20130322 17:44:49 
Last modified on  20130322 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 uppersemilattice 
Synonym  countably complete lowersemilattice 
Synonym  complete uppersemilattice homomorphism 
Synonym  complete lowersemilattice homomorphism 
Related topic  CompleteLattice 
Related topic  Semilattice 
Related topic  ArbitraryJoin 
Defines  countably complete joinsemilattice 
Defines  countably complete meetsemilattice 
Defines  complete joinsemilattice homomorphism 
Defines  complete meetsemilattice homomorphism 