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We firstly review some fundamentals of fuzzy relation calculus and, by recalling some known results, we improve the mathematical contents of our previous papers by using the properties of a triangular norm over [0, 1]. We make wide use of the theory of fuzzy relation equations for getting lossy compression and decompression of images interpreted as two-argument fuzzy matrices. The same scope is achieved by decomposing a fuzzy matrix using the concept of Schein rank. We illustrate two algorithms with a few examples. [ABSTRACT FROM AUTHOR]

Distributivity between two operations is a property posed many years ago — that is especially interesting in the framework of logical connectives because of its applications to fuzzy logic and approximate reasoning as their applications. Since semi-uninorms have been used in these topics, the study of the distributivity between two semi-uninorms becomes of particular interest that calls for thorough studies. The distributivity between two semi-uninorms, which are non-commutative and non-associative uninorms, has been developed only in the cases when both semi-uninorms are examples of very special classes of semi-uninorms. On the other hand, in general, the distributivity does not rely on the commutativity and associativity. The objective of this work is twofold. The first one is to show new solutions to distributivity equations for semi-uninorms. The second one is to check whether the results concerning the distributivity between two uninorms are valid for semi-uninorms. We investigate the distributivity involving two semi-uninorms when only one semi-uninrom lies in the most studied classes of semi-uninorms, achieving the above two objectives simultaneously. [ABSTRACT FROM AUTHOR]