Cancellative residuated lattices |
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Authors: | Email author" target="_blank">P?BahlsEmail author J?Cole N?Galatos P?Jipsen C?Tsinakis |
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Institution: | (1) Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, 37240 Nashville, TN, USA |
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Abstract: | Cancellative residuated lattices are natural generalizations of lattice-ordered
groups (
-groups).
Although cancellative monoids are defined by quasi-equations, the class
of cancellative residuated lattices is a variety.
We prove that there are only two
commutative subvarieties of
that cover the trivial variety, namely the varieties
generated by the integers and the negative integers (with zero). We also construct examples
showing that in contrast to
-groups, the lattice reducts of cancellative residuated lattices
need not be distributive. In fact we prove that every lattice can be embedded in the
lattice reduct of a cancellative residuated lattice. Moreover, we show that there exists an
order-preserving injection of the lattice of all lattice varieties into the subvariety lattice of
.We define generalized MV-algebras and generalized BL-algebras and prove that the
cancellative integral members of these varieties are precisely the negative cones of
-groups, hence the latter form a variety, denoted by
. Furthermore we prove that the map that sends a subvariety of
-groups to the corresponding class of negative cones is a lattice
isomorphism from the lattice of subvarieties of
to the lattice of subvarieties of
.
Finally, we show how to translate equational bases between corresponding subvarieties, and
briefly discuss these results in the context of R. McKenzies characterization of categorically
equivalent varieties. |
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Keywords: | 06F05 06D35 06F15 08B15 |
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