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

Dvurečenskij, Anatolij; Rachůnek, Jiří

Discrete Math. 306, No. 13, 1317-1326 (2006).

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 \(\mathbf{A}\).

Reviewer: Rostislav Horcik (Praha)

MSC:

06F05	Ordered semigroups and monoids
06D35	MV-algebras

Keywords:

bounded commutative Rl-monoid; Bosbach state; extremal state; maximal filter; MV-algebra; BL-algebra

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