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complete Boolean algebra
A Boolean algebra $A$ is a complete Boolean algebra if for every subset $C$ of $A$, the arbitrary join and arbitrary meet of $C$ exist.
By de Morgan’s laws, it is easy to see that a Boolean algebra is complete iff the arbitrary join of any subset exists iff the arbitrary meet of any subset exists. For a proof of this, see this link.
For an example of a complete Boolean algebra, let $S$ be any set. Then the powerset $P(S)$ with the usual set theoretic operations is a complete Boolean algebra.
In a complete Boolean algebra, the infinite distributive and infinite deMorgan’s laws hold:

$x\wedge\bigvee A=\bigvee(x\wedge A)$ and $x\vee\bigwedge A=\bigwedge(x\vee A)$

$(\bigvee A)^{*}=\bigwedge A^{*}$ and $(\bigwedge A)^{*}=\bigvee A^{*}$, where $A^{*}:=\{a^{*}\mid a\in A\}$.
In the category of complete Boolean algebras, a morphism between two objects is a Boolean algebra homomorphism that preserves arbitrary joins (equivalently, arbitrary meets), and is called a complete Boolean algebra homomorphism.
Remark There are infinitely many algebras between Boolean algebras and complete Boolean algebras. Let $\kappa$ be a cardinal. A Boolean algebra $A$ is said to be $\kappa$complete if for every subset $C$ of $A$ with $C\leq\kappa$, $\bigvee C$ (and equivalently $\bigwedge C$) exists. A $\kappa$complete Boolean algebra is usually called a $\kappa$algebra. If $\kappa=\aleph_{0}$, the first aleph number, then it is called a countably complete Boolean algebra.
Any complete Boolean algebra is $\kappa$complete, and any $\kappa$complete is $\lambda$complete for any $\lambda\leq\kappa$. An example of a $\kappa$complete algebra that is not complete, take a set $S$ with $\kappa<S$, then the collection $A\subseteq P(S)$ consisting of any subset $T$ such that either $T\leq\kappa$ or $ST\leq\kappa$ is $\kappa$complete but not complete.
A Boolean algebra homomorphism $f$ between two $\kappa$algebras $A,B$ is said to be $\kappa$complete if
$f(\bigvee\{a\mid a\in C\})=\bigvee\{f(a)\mid a\in C\}$ 
for any $C\subseteq A$ with $C\leq\kappa$.
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