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Prime ideals and polars in DR\(\ell \)-monoids and BL-algebras. (English) Zbl 1058.06017
Dually residuated lattice-ordered monoids (DR\(\ell\)-monoids) form a wide class of algebras which contains, e.g., the classes of lattice-ordered groups (\(\ell \)-groups) and classes of algebras behind fuzzy reasoning like MV-algebras, pseudo MV-algebras (GMV-algebras), BL-algebras or pseudo BL-algebras. The author studies structure properties of DR\(\ell \)-monoids that are in connection with prime ideals, i.e., the prime elements of the lattices of ideals of DR\(\ell \)-monoids. Using a simple condition, he obtains results similar to those in the theory of \(\ell \)-groups. Since all dual pseudo BL-algebras satisfy this condition, the main results are reformulated for the theory of pseudo BL-algebras.

MSC:
06F05 Ordered semigroups and monoids
06D35 MV-algebras
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References:
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