  Given a (finite or infinite) subset $X$ of the free monoid $A^*$ over a
finite alphabet $A$, the rank of $X$ is the minimal cardinality of a set $F$
such that $X \subseteq F^*$. We say that a submonoid $M$ generated by $k$
elements of $A^*$ is {\em $k$-maximal} if there does not exist another
submonoid generated by at most $k$ words containing $M$. We call a set $X
\subseteq A^*$ {\em primitive} if it is the basis of a $|X|$-maximal submonoid.
This definition encompasses the notion of primitive word -- in fact, $\{w\}$ is
a primitive set if and only if $w$ is a primitive word. By definition, for any
set $X$, there exists a primitive set $Y$ such that $X \subseteq Y^*$. We
therefore call $Y$ a {\em primitive root} of $X$. As a main result, we prove
that if a set has rank $2$, then it has a unique primitive root. To obtain this
result, we prove that the intersection of two $2$-maximal submonoids is either
the empty word or a submonoid generated by one single primitive word. For a
single word $w$, we say that the set $\{x,y\}$ is a {\em bi-root} of $w$ if $w$
can be written as a concatenation of copies of $x$ and $y$ and $\{x,y\}$ is a
primitive set. We prove that every primitive word $w$ has at most one bi-root
$\{x,y\}$ such that $|x|+|y|<\sqrt{|w|}$. That is, the bi-root of a word is
unique provided the word is sufficiently long with respect to the size (sum of
lengths) of the root. Our results are also compared to previous approaches that
investigate pseudo-repetitions, where a morphic involutive function $\theta$ is
defined on $A^*$. In this setting, the notions of $\theta$-power,
$\theta$-primitive and $\theta$-root are defined, and it is shown that any word
has a unique $\theta$-primitive root. This result can be obtained with our
approach by showing that a word $w$ is $\theta$-primitive if and only if $\{w,
\theta(w)\}$ is a primitive set.
