Functions, Maps & Integral Structure

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Abstractions about functions and their graphs, images and kernels, monotonicity, convexity, integrals, mapping properties, and model transformations.

10 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Daniell Integral — A function-first integration construction that extends a positive monotone-continuous linear functional from an elementary function lattice and derives measure afterward.
• Graph of a Function — Represent a function by the set of ordered input–output pairs selected by its evaluation rule, preserving every domain element with exactly one associated value while separating the graph from a plotted picture or graph-theoretic network.
• Image (of a Function) — The set of values a function actually produces — im(f) = {f(x) : x in X} — a subset of the declared codomain that records what a map reaches rather than what it could in principle reach.
• Kernel — The set of inputs a structure-preserving map sends to the identity of its target — the map's null directions — collected as a genuine subobject that carries the map's information loss and, via triviality, rank-nullity, and the isomorphism theorem, controls injectivity and reconstructs the faithful part.
• MOF Model to Text Transformation Language — Specify repeatable model-to-text generation with typed MOF-model inputs, guarded templates, queries, iteration, file routing, trace links, and protected regions for generated code or documentation.
• Monotonic Function — Map one ordered set into another while preserving comparison direction everywhere, or reverse that direction everywhere in the antitone variant, so input order constrains output order globally.
• Quasi-Open Map — Map every nonempty open subset of a topological domain to a set with nonempty interior, retaining a weakened openness guarantee without requiring the image itself to be open.
• Quasiconvex Function — A real-valued function on a convex domain whose sublevel sets are convex, preserving convex feasibility under every threshold without requiring the stronger convex-function inequality.
• Schröder–Bernstein Property — The property of a specified mathematical class, embedding relation, and equivalence notion that mutual embeddability forces equivalence, making the embedding preorder antisymmetric after quotienting by that equivalence.
• Unisolvent Point Set — A finite sampling set for which evaluation on a declared finite-dimensional function space is injective, equivalently making interpolation uniquely solvable when the dimensions match.