Pointed lattice subreducts of varieties of residuated lattices

Abstract: We study the pointed lattice subreducts of varieties of residuated lattices (RLs) and commutative residuated lattices (CRLs), i.e. lattice subreducts expanded by the constant 1 denoting the multiplicative unit. Given any positive universal class of pointed lattices K satisfying a certain equation, we describe the pointed lattice subreducts of semi-K and of pre-K RLs and CRLs. The quasivariety of semi-prime-pointed lattices generated by pointed lattices with a join prime constant 1 plays an important role here. In particular, the pointed lattices reducts of integral (semiconic) RLs and CRLs are precisely the integral (semiconic) semi-prime-pointed lattices. We also describe the pointed lattice subreducts of integral cancellative CRLs, proving in \mbox{particular} that every lattice is a subreduct of some integral cancellative CRL. This resolves an open problem about cancellative CRLs.