Generalized Lattices Express Parallel Distributed Concept Learning

Concepts have been expressed mathematically as propositions in a distributive lattice. A more comprehensive formulation is that of a generalized lattice, or category, in which the concepts are related in hierarchical fashion by lattice-like links called concept morphisms. A concept morphism describes how a more abstract concept is used within a more specialized concept, as the color "red" is used in describing "apples". Often, an abstract concept can be used in a more specialized concept in more than one way as with "color", which can appear in "apples" as either "red", "yellow" or "green". Further, "color" appears in "apples" because it appears in "red", "yellow" and "green", which in turn appear in "apples", expressed via the composition of concept morphisms. Using categorical constructs based upon composition together with structure-preserving mappings that preserve compositional structure, a recently-developed semantic theory shows how abstract and specialized concepts are learned by a neural network.