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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
##### Keywords:
DR$$\ell$$-monoid; pseudo BL-algebra; prime ideal; polar
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##### References:
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