Subdirectly Irreducible complex c-PBL-algebras

CORDERO, PENÉLOPE

Concepción

Simposio; XVIII Simposio Latinoamericano de Lógica Matemática SLALM 2019; 2019

Universidad de Concepción

Because of the success of the propositional system BL [4], the modal fuzzy logicswhose semantics is given by frames over BL-algebras (the algebraic counterpart of BL)are an interesting problem addressed by many people. Particularly, for the case ofthe fuzzy analogue of the modal system KD45, unlike the classical cases, the standardmethods to give an axiomatization fail. For this reason, we take an algebraic approachdefining the class of Pseudomonadic BL-algebras (PBL-algebras) [2].Considering the previous related works, we show that Pseudomonadic Boolean algebras coincide with the algebraic counterpart of classical KD45 [1], while the G¨odelPBL-algebras correspond with the class of bi-modal G¨odel algebras studied in [3].On the other hand, we prove that given a possibilistic BL-frame, its associated complex algebra is a special case of PBL-algebra (c-PBL-algebra). Thus, the complexc-PBL-algebras allow to establish a connection between relational and algebraic semantics. Since our aim is to show that the class of PBL-algebras is equivalent to thesemantics given by Kripke BL-frames, we focus on the study of complex c-PBL-algebras.We characterize the subdirectly irreducible elements in this class.[1] Bezhanishvili, N., Pseudomonadic algebras as algebraic models of doxasticmodal logic, MLQ Math. Log. Q., vol. 48 (2002), no. 4, pp. 624?636.[2] Busaniche, M., Cordero,P., Rodriguez, R.O., Pseudomonadic BL-algebras:an algebraic approach to possibilistic BL-logic, Soft Computing, vol. 23 (2019), no. 7,pp. 2199?2212.[3] Caicedo, X., Rodriguez, R.O., Bi-modal G¨odel logic over [0, 1]-valued Kripkeframes, J. Logic Comput., vol. 25 (2015), no. 1, pp. 37?55.[4] Hajek, P. ´ , Metamathematics of fuzzy logic, Trends in Logic Studia LogicaLibrary, Kluwer Academic Publishers, Dordrecht, 1998