NDLI: Cancellative residuated lattices

Content Provider	Springer Nature Link
Author	Bahls, P. Galatos, N. Jipsen, P. Tsinakis, C. Cole, J.
Copyright Year	2003
Abstract	Cancellative residuated lattices are natural generalizations of lattice-ordered groups ( $$ \mathcal{l} $$ -groups). Although cancellative monoids are defined by quasi-equations, the class $$ \mathcal{CanRL} $$ of cancellative residuated lattices is a variety. We prove that there are only two commutative subvarieties of $$ \mathcal{CanRL} $$ 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 $$ \mathcal{l} $$ -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 $$ \mathcal{CanRL} $$ .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 $$ \mathcal{l} $$ -groups, hence the latter form a variety, denoted by $$ \mathcal{LG}^- $$ . Furthermore we prove that the map that sends a subvariety of $$ \mathcal{l} $$ -groups to the corresponding class of negative cones is a lattice isomorphism from the lattice of subvarieties of $$ \mathcal{LG}$$ to the lattice of subvarieties of $$ \mathcal{LG}^- $$ . Finally, we show how to translate equational bases between corresponding subvarieties, and briefly discuss these results in the context of R. McKenzie’s characterization of categorically equivalent varieties.
Ending Page	106
Page Count	24
Starting Page	83
File Format	PDF
ISSN	00025240
e-ISSN	14208911
Journal	algebra universalis
Issue Number	1
Volume Number	50
Language	English
Publisher	Birkhäuser-Verlag
Publisher Date	2003-01-01
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Ordered semigroups and monoids Lattices of varieties MV-algebras Ordered groups
Content Type	Text
Resource Type	Article
Subject	Logic Algebra and Number Theory

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