  Monoids generated by elements of order two appear in numerous places in the
literature. For example, Coxeter reflection groups in geometry, Kuratowski
monoids in topology, various monoids generated by regular operations in
language theory and so on. In order to initiate a classification of these
monoids, we are interested in the subproblem of monoids, called strict 2-PIMs,
generated by an involution and an idempotent. In this case we show, when the
monoid is finite, that it is generated by a single equation (in addition to the
two defining the involution and the idempotent). We then describe the exact
possible forms of this equation and classify them. We recover Kuratowski's
theorem as a special case of our study.
