Abstract In this work, we provide constructions, characterizations, geometrical dualities and a McNaughton theorem for non-archimedean MV-algebras, which are the semantics of super-Łukasiewicz logics introduced by Komori.

Abstract We offer a new logic, called Epistemic Łukasiewicz logic of partial knowledge that is represented as multimodal epistemic Łukasiewicz logic $$K\text{\L} _\text {P}(n)$$ K Ł P ( n ) with n knowledge operators $$\Box _i$$ □ i $$(1 \le i \le n)$$ ( 1 ≤ i ≤ n ) interpreted in a non-archimedean monadic MV-algebra. We choose knowledge operators, which can be estimated by some grading (different kinds of knowledge): absolute knowledge or partial knowledge. We consider a very special type of partial knowledge. Actually, we take infinitesimal elements (the radical) of perfect MV-algebras as a range of this estimation. The choice of infinitesimal elements seems suitable for actual situations like a measure of partial information.

Abstract It is introduced an immune dynamicn-valued Łukasiewicz logic $$ID{\L }_n$$ IDŁn on the base ofn-valued Łukasiewicz logic $${\L }_n$$ Łn and corresponding to it immune dynamic $$MV_n$$ MVn -algebra ( $$IDL_n$$ IDLn -algebra), $$1< n < \omega $$ 1<n<ω , which are algebraic counterparts of the logic, that in turn represent two-sorted algebras $$(\mathcal {M}, \mathcal {R}, \Diamond )$$ (M,R,◊) that combine the varieties of $$MV_n$$ MVn -algebras $$\mathcal {M} = (M, \oplus , \odot , \sim , 0,1)$$ M=(M,⊕,⊙,∼,0,1) and regular algebras $$\mathcal {R} = (R,\cup , ;, ^*)$$ R=(R,∪,;,∗) into a single finitely axiomatized variety resemblingR-module with “scalar” multiplication $$\Diamond $$ ◊ . Kripke semantics is developed for immune dynamic Łukasiewicz logic $$ID{\L }_n$$ IDŁn with application in immune system.

We focus on Riesz MV-algebras, which are MV-algebras equipped with a multiplication by numbers in the real interval [0,1]. In analogy with a work in preparation for MV-algebras by the same authors, we consider for every integer n the Riesz MV-algebra of all continuous functions from the n-th power of [0,1] to [0,1] and the Riesz MV-subalgebras thereof. In particular we study the Riesz MV-subalgebras isomorphic to free Riesz MV-algebras with finitely many generators, possibly different from the usual linear models given by what we call Riesz-McNaughton functions (and which generalize McNaughton functions used in the case of MV-algebras). In doing this we characterise zerosets of Riesz-McNaughton functions by means of polyhedra, and we extend to Riesz MV-algebras a duality for MV-algebras exposed in a paper by Marra and Spada.