Completeness for monadic fuzzy logics via functional algebras
Section snippets
Introduction and preliminaries
We denote by the basic logic defined by Hájek in [18]. is an example of an implicative logic, so it is an algebraizable logic whose equivalent algebraic semantics is the variety of BL-algebras that we usually denote by . As a result of the general theory of algebraizability, every axiomatic extension of is also algebraizable and its equivalent algebraic semantics is a subvariety of . More precisely, if is an axiomatic extension of , the equivalent algebraic semantics
Completeness via functional algebras
In this section we define -functional algebras and prove that, given an axiomatic extension of , the completeness theorem holds, that is, , if and only if the variety is generated as a quasivariety by its -functional algebras.
To define -functional algebras, we need to recall how to define monadic BL-algebras based on m-relatively complete subalgebras of a BL-algebra. In [10] we give a characterization of those subalgebras of a given BL-algebra that may be the range of
Monadic Łukasiewicz logic
One of the main axiomatic extensions of basic logic is Łukasiewicz logic , which may be obtained by adding the axiom schema to . Its corresponding equivalent algebraic semantics is the variety of MV-algebras, which is precisely the subvariety of determined by the identity . We refer the reader to [11] for all basic properties of MV-algebras.
We recall here some useful notions of BL-algebras and MV-algebras that will be needed. Given a BL-algebra A, the congruences on A are in
Monadic Gödel logic
Another important axiomatic extension of is the one obtained by adding the axiom . This is usually called Gödel logic and denoted by . The corresponding equivalent algebraic semantics is the variety of Gödel algebras, that is, the subvariety of determined by the identity (or, equivalently, by the identity ). We denote this variety by . In addition, Gödel algebras may also be defined as prelinear Heyting algebras, that is, is the variety generated by totally ordered
Concluding remarks and future work
In this article we presented a way of proving the completeness of certain monadic logical calculi by means of algebraic representation theorems. We would like to pursue this route even deeper in order to prove the completeness for monadic basic logic. The key fact we used to prove the representation theorems in the cases addressed in this article was the amalgamation property. However, this is not enough in general. We already have some partial results that point in the right direction, but
Acknowledgements
We would like to thank the anonymous referees for their useful remarks that improved the clarity of this article. We would also like to acknowledge the funding received by Universidad Nacional del Sur (grant number 24/L108) and CONICET (grant number 11220170101010CO).
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