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 pseudo-linear triangle logic (PTL), an axiomatic extension of TL introduced in this paper. This kind of completeness is the analogue of the chain completeness of monoidal T-norm based logic (MTL). 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 image, in which image is a residuated lattice and image an element in that residuated lattice. We give a schematic overview of some properties of pseudo-prelinear triangle algebras (and a number of others that can be imposed on a triangle algebra), and the according necessary and sufficient conditions on image and image.