The semilattice relevant logics ~∪R, ~∪T, ~∪RW, and ~∪TW (slightly different from the orthodox relevant logics R, T, RW, and TW) are defined by semilattice models in which conjunction and disjunction are interpreted in a natural way. For each of them, there is a cut-free labelled sequent calculus with plural succedents (like LK). We prove that these systems are equivalent, with respect to provable formulas, to the restricted systems with single succedents (like LJ). Moreover, using this equivalence, we give a new Hilbert-style axiomatizations for ~∪R and ~∪T and prove equivalence between two semantics (commutative monoid and distributive semilattice) for the contractionless logics ~∪RW and ~∪TW.
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