The logical content of triangular bases of fuzzy sets in {\L}ukasiewicz infinite-valued logic

Continuing to pursue a research direction that we already explored in connection with G"odel-Dummett logic and Ruspini partitions, we show here that {\L}ukasiewicz logic is able to express the notion of pseudo-triangular basis of fuzzy sets, a mild weakening of the standard notion of triangular basis. En route to our main result we obtain an elementary, logic-independent characterisation of triangular bases of fuzzy sets.