  In this paper we define future-time branching temporal logics evaluated over
forests, that is, ordered tuples of ordered, but unranked, finite trees. We
associate a rich class FL[$\mathcal{L}$] of temporal logics to each set L of
(regular) modalities. Then, we define an algebraic product operation which we
call the Moore product, which operates on forest automata, algebraic devices
recognizing forest languages. We show a lattice isomorphism between the
pseudovarieties of finite forest automata, closed under the Moore product, and
the classes of languages of the form FL[$\mathcal{L}$]. We demonstrate the
usefulness of the algebraic approach by showing the decidability of the
membership problem of a specific pseudovariety of finite forest automata,
implying the decidability of the definability problem of the FL[EF] fragment of
the logic CTL. Then, using the same approach, we also formulate a conjecture
regarding a decidable characterization of the FL[AF] fragment which has
currently an unknown decidability status (also in the setting of ranked trees).
