The pseudo-linear semantics of interval-valued fuzzy logics

Triangle algebras are equationally defined structures that are equivalent with certain residuated lattices on a set of intervals, which are called interval-valued residuated lattices (IVRLs). Triangle algebras have been used to construct Triangle Logic (TL), a formal fuzzy logic that is sound and complete w.r.t. the class of IVRLs. In this paper, we prove that the so-called pseudo-prelinear triangle algebras are subdirect products of pseudo-linear triangle algebras. This can be compared with MTL-algebras (prelinear residuated lattices) being subdirect products of linear residuated lattices. As a consequence, we are able to prove the pseudo-chain completeness of Pseudolinear Triangle Logic (PTL), an axiomatic extension of TL introduced in this paper. This kind of completeness is the analogue of the chain completeness of MTL (Monoidal T-norm based Logic). This result also provides a better insight in the structure of triangle algebras; it enables us, amongst others, to prove properties of pseudo-prelinear triangle algebras more easily. It is known that there is a one-to-one correspondence between triangle algebras and couples (L, α), in which L is a residuated lattice and α an element in that residuated lattice. We give a schematic overview of these properties (and a number of others that can be imposed on a triangle algebra), and the corresponding necessary and sufficient conditions on L and α.