Every Linear Pseudo BL-Algebra Admits a State

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. The paper has been supported by the Center of Excellence SAS—Physics of Information—I/2/2005, the grant VEGA no. 2/3163/23 SAV and by Science and Technology Assistance Agency under the contract no. APVT-51-032002. Bratislava, Slovakia.