Every Linear Pseudo BL-Algebra Admits a State

Authors:
Anatolij Dvurečenskij
Affiliations:
Slovak Academy of Sciences, Mathematical Institute, Štefánikova 49, 814 73, Bratislava, Slovakia
Venue:
Soft Computing - A Fusion of Foundations, Methodologies and Applications
Year:
2007

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Abstract

We show that every linear pseudo BL-algebra, hence every representable one, admits a state and is good. This solves positively the problem on the existence of states raised in Dvurečenskij and Rachůnek (Probabilistic averaging in bounded communitative residuated ℓ-monoids, 2006), and gives a partial answer to the problem on good pseudo BL-algebras from [Di Nola, Georgescu and Iorgulescu (Multiple Val Logic 8:715–750, 2002) Problem 3.21]. Moreover, we present that every saturated linear pseudo BL-algebra can be expressed as an ordinal sum of Hájek’s type of irreducible pseudo linear pseudo BL-algebras.