Characterization and Continuity of Fuzzy Morphological Associative Memories on Complete Lattice-Ordered Double Monoids

Fuzzy associative memories (FAMs) are between-cube fuzzy system. They are often defined as artificial neural networks whose inputs, outputs, and connection weights are fuzzy valued. Recently, Valle and Sussner observed that many FAM models are equipped with neurons that perform elementary operations of mathematical morphology such as dilation or erosion. Thus, they can be classified as fuzzy morphological associative memories (FMAMs). Although complete lattices provide a general framework for FMAMs, in this paper we note that these models can be completely characterized in a mathematical structure called clodum or complete lattice-ordered double monoid. Precisely, in a clodum, a between-cube fuzzy system and consequently an FMAM model can be viewed a fuzzy logic neural network if, and only if, it yields a mapping that performs either a dilation or an erosion that is invariant under a certain type of membership regraduations. Furthermore, we show that fuzzy learning by adjunction yields a continuous FMAM if all the association pairs are memorized correctly.