## Probabilistic averaging in bounded commutative residuated $$\ell$$-monoids.(English)Zbl 1105.06011

The paper deals with a characterization of Bosbach states on bounded commutative residuated $$l$$-monoids. This class of algebras can be obtained from BL-algebras by omitting the defining identity of prelinearity (i.e. $$(x\to y)\vee (y\to x)\approx 1$$). The paper in fact generalizes two basic results on states, which are known for MV-algebras, to the case of the above-mentioned algebras. The first one says that, given a bounded commutative residuated $$l$$-monoid $$\mathbf{A}$$, the set of all states on $$\mathbf{A}$$ forms a non-empty compact convex Hausdorff space with respect to the weak topology, and the set of all state-morphisms on $$\mathbf{A}$$ is a non-empty compact Hausdorff space. The second result shows that the set of all maximal filters of $$\mathbf{A}$$ endowed with the hull-kernel topology is homeomorphic to the space of all state-morphisms (i.e. extremal states) on $$\mathbf{A}$$.

### MSC:

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

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