  We introduce a family of modal expansions of {\L}ukasiewicz logic that are
designed to accommodate modal translations of generalized basic logic (as
formulated with exchange, weakening, and falsum). We further exhibit algebraic
semantics for each logic in this family, in particular showing that all of them
are algebraizable in the sense of Blok and Pigozzi. Using this algebraization
result and an analysis of congruences in the pertinent varieties, we establish
that each of the introduced modal {\L}ukasiewicz logics has a local
deduction-detachment theorem. By applying Jipsen and Montagna's poset product
construction, we give two translations of generalized basic logic with
exchange, weakening, and falsum in the style of the celebrated
G\"odel-McKinsey-Tarski translation. The first of these interprets generalized
basic logic in a modal {\L}ukasiewicz logic in the spirit of the classical
modal logic S4, whereas the second interprets generalized basic logic in a
temporal variant of the latter.
