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Dually residuated \(\ell\)-monoids having no non-trivial convex subalgebras. (English) Zbl 1125.06012

The paper gives a characterization of simple dually residuated \(\ell\)-monoids (DR\(\ell\)-monoids), i.e. DR\(\ell\)-monoids containing only the trivial convex subalgebra. Based on the known fact that each DR\(\ell\)-monoid is isomorphic to a direct product of an \(\ell\)-group and an lower bounded DR\(\ell\)-monoid, the author proves that a simple DR\(\ell\)-monoid is either an Archimedean totally ordered group or an Archimedean totally ordered generalized pseudo MV-algebra. Consequently, each such DR\(\ell\)-monoid has to be commutative.

MSC:

06F05 Ordered semigroups and monoids

Software:

Pseudo Hoops
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References:

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