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Finite-valued dually residuated lattice-ordered monoids. (English) Zbl 1141.06014
Summary: Lattice-ordered groups, as well as \(\roman {GMV}\)-algebras (called also pseudo \(\roman {MV}\)-algebras), are both particular cases of dually residuated lattice-ordered monoids (\(\roman {DR}\ell \)-monoids). In the paper we study values in \(\roman {DR}\ell \)-monoids, especially if the ideal lattice is a member of the class \(\mathcal {TRN}\) of algebraic, distributive lattices whose compact elements form a relatively normal sublattice, and we characterize finite-valued \(\roman {DR}\ell \)-monoids whose ideal lattices belong to \(\mathcal {TRN}\).

MSC:
06F05 Ordered semigroups and monoids
06D35 MV-algebras
03G25 Other algebras related to logic
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