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 \(\textsf{1}\) denoting the multiplicative unit. Given any positive universal class of pointed lattices \(\textsf{K}\) satisfying a certain equation, we describe the pointed lattice subreducts of semi-\(\textsf{K}\) and of pre-\(\textsf{K}\) RLs and CRLs. The quasivariety of semi-prime-pointed lattices generated by pointed lattices with a join prime constant \(\textsf{1}\) plays an important role here. In particular, the pointed lattice 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 particular that every lattice is a subreduct of some integral cancellative CRL. This resolves an open problem about cancellative CRLs.