A residuated semilattice is defined almost identically for residuated lattices, omitting just the meet operation ∧. Thus it is an algebraic structure L = (L, ∨, •, 1, /, \) satisfying all the residuated lattice equations as specified above except those containing an occurrence of the symbol ∧. The option of defining x ≤ y as x∧y = x is then not available, leaving only the other option x∨y = y (or any equivalent thereof).
Any residuated lattice can be made a residuated semilattice simply by omitting ∧. Residuated semilattices arise in connection with action algebras, which are residuated semilattices that are also Kleene algebras, for which ∧ is ordinarily not required.
Read more about this topic: Residuated Lattice