  Quasi-set theory was proposed as a mathematical context to investigate
collections of indistinguishable objects. After presenting an outline of this
theory, we define an algebra that has most of the standard properties of an
orthocomplete orthomodular lattice, which is the lattice of the closed
subspaces of a Hilbert space. We call the mathematical structure so obtained
$\mathfrak{I}$-lattice. After discussing, in a preliminary form, some aspects
of such a structure, we indicate the next problem of axiomatizing the
corresponding logic, that is, a logic which has $\mathfrak{I}$-lattices as its
algebraic models. We suggest that the intuitions that the `logic of quantum
mechanics' would be not classical logic (with its Boolean algebra), is
consonant with the idea of considering indistinguishability right from the
start, that is, as a primitive concept. In other words, indiscernibility seems
to lead `directly' to $\mathfrak{I}$-lattices. In the first sections, we
present the main motivations and a `classical' situation which mirrors that one
we focus on the last part of the paper. This paper is our first study of the
algebraic structure of indiscernibility within quasi-set theory.
