Sequences, Mappings & Basepoint Structures

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Abstractions that loosely connect mathematical constructions built from mappings and sequences, including function classification by iterative construction such as Baire functions, low-discrepancy and registration methods like the Halton sequence and point-set registration, and structure-preserving mappings such as inclusion maps, point reflection and tree rotation.

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.

• Baire function — A real-valued function obtainable from continuous functions by countable transfinite iteration of pointwise sequential limits, classified by the least countable ordinal stage required.
• FOSD program cubes — An n-dimensional feature-oriented representation whose axes encode independent variability dimensions and whose cells or transformations compose into members of a software product line.
• Halton sequence — A deterministic low-discrepancy sequence in the unit cube formed by combining one-dimensional radical-inverse sequences in pairwise coprime bases.
• Inclusion map — The canonical injective function from a subset or subobject into its containing object that sends every element to itself viewed in the larger context.
• Point reflection — An affine transformation that sends each point x to 2c−x about a fixed center c, preserving distances and reversing every displacement vector.
• Point-set registration — The estimation of a spatial transformation that aligns two or more point sets into a common coordinate frame despite noise, outliers or incomplete overlap.
• Pointed set — A set equipped with one distinguished basepoint, with morphisms required to preserve that point, forming a category that adds a canonical zero-like reference to otherwise unstructured sets.
• Semigroupoid — A category-like partial algebra with objects, composable morphisms and associative composition but without requiring an identity morphism at every object.
• Transport of Structure — Define operations, relations, or other structure on one mathematical carrier through a chosen equivalence so that the equivalence becomes structure-preserving by construction.
• Tree rotation — A local binary-tree restructuring that preserves in-order key order while changing parent-child shape.