Semiring and Semimodule Issues in MV-Algebras

Module Structure on Effect Algebras

“…The monograph [2,11] can serve as a basic source of information about effect algebras. Product effect algebras, were introduced by Anatolij Dvurecenskij [12]. He proved every product effect algebra with the Riesz decomposition property (RDP ) is an interval in an Abelian unital interpolation po-ring, and he showed that the category of product effect algebras with the RDP is categorically equivalent with the category of unital Abelian interpolation po-rings.…”

“…He proved every product effect algebra with the Riesz decomposition property (RDP ) is an interval in an Abelian unital interpolation po-ring, and he showed that the category of product effect algebras with the RDP is categorically equivalent with the category of unital Abelian interpolation po-rings. Recently, some researchers worked on modular structures (see, for instance, [3,4,9,10,17]). Effect modules have been introduced in theoretical physics in the mid-1990 for quantum probability.…”

Quantales and their modules: projective objects, ideals, and congruences

“…Then, using such a result, we shall characterize projective modules by means of multiplicatively idempotent elements of Q X×X (Theorem 2.4). We will conclude Section 2 by observing that the construction of the K 0 group of a semiring, presented in [8], can be plainly applied to quantales, thus promisingly broadening this topic's horizons.…”

Tropical linear algebra with the Łukasiewicz T-norm

“…There are directions of abstract fuzzy sets theory which are related to the present work. Some connections of multivalued logic and fuzzy algebra with idempotent mathematics have been developed by Di Nola et al [18,19,20,21]. These works develop certain aspects of algebra over semirings arising from fuzzy logic (MV-algebras, Lukasiewicz transform), which currently seem most interesting and useful for the fuzzy sets theory.…”