Abstract
AbstractWe study the pointed lattice subreducts of varieties of residuated lattices (RLs) and commutative residuated lattices (CRLs), i.e. lattice subreducts expanded by the constant $$\textsf{1}$$
1
denoting the multiplicative unit. Given any positive universal class of pointed lattices $$\textsf{K}$$
K
satisfying a certain equation, we describe the pointed lattice subreducts of semi-$$\textsf{K}$$
K
and of pre-$$\textsf{K}$$
K
RLs and CRLs. The quasivariety of semi-prime-pointed lattices generated by pointed lattices with a join prime constant $$\textsf{1}$$
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.
Publisher
Springer Science and Business Media LLC
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