  This article fits in the area of research that investigates the application
of topological duality methods to problems that appear in theoretical computer
science. One of the eventual goals of this approach is to derive results in
computational complexity theory by studying appropriate topological objects
which characterize them. The link which relates these two seemingly separated
fields is logic, more precisely a subdomain of finite model theory known as
logic on words. It allows for a description of complexity classes as certain
families of languages, possibly non-regular, on a finite alphabet. Very few is
known about the duality theory relative to fragments of first-order logic on
words which lie outside of the scope of regular languages. The contribution of
our work is a detailed study of such a fragment. Fixing an integer $k \geq 1$,
we consider the Boolean algebra $\mathcal{B}\Sigma_1[\mathcal{N}^{u}_k]$. It
corresponds to the fragment of logic on words consisting in Boolean
combinations of sentences defined by using a block of at most $k$ existential
quantifiers, letter predicates and uniform numerical predicates of arity $l \in
\{1,...,k\}$. We give a detailed study of the dual space of this Boolean
algebra, for any $k \geq 1$, and provide several characterizations of its
points. In the particular case where $k=1$, we are able to construct a family
of ultrafilter equations which characterize the Boolean algebra $\mathcal{B}
\Sigma_1[\mathcal{N}^{u}_1]$. We use topological methods in order to prove that
these equations are sound and complete with respect to the Boolean algebra we
mentioned.
