Mathematical Structure

A Mathematical Structure is a mathematical object consisting of a set together with mathematical operations and mathematical relations that provides semantic interpretation for formal mathematical systems.

• AKA: Algebraic Structure, Relational Structure, Model-Theoretic Structure.
• Context:
  • It can typically consist of a carrier set equipped with mathematical operations and mathematical relations.
  • It can typically satisfy mathematical axioms through property instantiation.
  • It can typically provide semantic interpretations for formal mathematical symbols.
  • It can typically exhibit mathematical properties through relational configurations.
  • It can typically support mathematical operations through function definitions.
  • It can typically enable truth evaluation for mathematical statements within its domain.
  • It can typically serve as model instances for formal mathematical systems.
  • It can typically define mathematical universes for model instantiation.
  • It can often model multiple formal mathematical systems through interpretation mappings.
  • It can often possess invariant properties under structure-preserving transformations.
  • It can often participate in category-theoretic relationships through mathematical morphisms.
  • It can often admit automorphism groups preserving structural features.
  • It can often provide semantic domains for metamodel interpretation.
  • It can often establish mathematical constraints for model families.
  • It can often be combined through mathematical constructions like products and coproducts.
  • It can range from being a Finite Mathematical Structure to being an Infinite Mathematical Structure, depending on its carrier set cardinality.
  • It can range from being a Simple Mathematical Structure to being a Complex Mathematical Structure, depending on its operation and relation count.
  • It can range from being a Discrete Mathematical Structure to being a Continuous Mathematical Structure, depending on its topological properties.
  • It can range from being a First-Order Mathematical Structure to being a Higher-Order Mathematical Structure, depending on its logical complexity.
  • It can range from being a Total Mathematical Structure to being a Partial Mathematical Structure, depending on its operation definedness.
  • It can be analyzed through model theory and universal algebra.
  • It can be compared via homomorphisms, isomorphisms, and embeddings.
  • It can be classified through structure theorems and representation theory.
  • ...
• Example(s):
  • Algebraic Mathematical Structures, such as:
    • Group Structures:
      • Integers Under Addition: (ℤ, +) with identity 0.
      • Symmetric Group: S_n with permutation composition.
      • Matrix Group: GL(n,ℝ) with matrix multiplication.
    • Ring Structures:
      • Integer Ring: (ℤ, +, ×) with two operations.
      • Polynomial Ring: ℝ[x] over real coefficients.
      • Matrix Ring: M_n(ℝ) with matrix operations.
    • Field Structures:
      • Real Number Field: (ℝ, +, ×) complete ordered field.
      • Complex Number Field: (ℂ, +, ×) algebraically closed.
      • Finite Field: GF(p^n) with prime power order.
  • Order Structures, such as:
    • Partially Ordered Sets:
      • Power Set with Inclusion: (P(S), ⊆).
      • Natural Numbers with Divisibility: (ℕ, |).
    • Lattice Structures:
      • Boolean Algebra: with meet, join, complement.
      • Subgroup Lattice: of group substructures.
  • Topological Structures, such as:
    • Metric Spaces:
      • Euclidean Space: (ℝⁿ, d) with standard metric.
      • Discrete Metric Space: with discrete topology.
    • Manifold Structures:
      • Differentiable Manifold: with smooth structure.
      • Riemannian Manifold: with metric tensor.
  • Relational Structures, such as:
    • Graph Structures: (V, E) with vertices and edges.
    • Database Relations: with tuple structures.
    • Equivalence Relations: defining partitions.
  • Model-Theoretic Structures, such as:
    • First-Order Structures: interpreting predicate logic.
    • Multi-Sorted Structures: with type systems.
    • Temporal Structures: with time ordering.
  • Hybrid Structures, such as:
    • Topological Groups: combining group and topology.
    • Ordered Fields: combining field and order.
    • Normed Vector Spaces: combining vector space and norm.
  • ...
• Counter-Example(s):
  • Formal Mathematical System, which provides syntactic framework rather than semantic content.
  • Mathematical Proof, which demonstrates logical derivation rather than structural property.
  • Mathematical Algorithm, which specifies computational procedure rather than static structure.
  • Mathematical Notation, which provides symbolic representation rather than semantic structure.
  • Mathematical Theorem, which states properties rather than defining structure.
• See: Mathematical Object (parent concept), Formal Mathematical System, Model Theory, Universal Algebra, Category Theory, Structure-Preserving Map, Mathematical Metamodel, Isomorphism, Homomorphism, Automorphism, Mathematical Model, Algebraic Structure, Relational Structure, Topological Structure, Mathematical Space, Mathematical Analysis Task, Abstract State Machine, Concrete Category.

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References

2014

• http://en.wikipedia.org/wiki/Mathematical_model
  • A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences (such as physics, biology, earth science, meteorology) and engineering disciplines (e.g. computer science, artificial intelligence), but also in the social sciences (such as economics, psychology, sociology and political science); physicists, engineers, statisticians, operations research analysts and economists use mathematical models most extensively. A model may help to explain a system and to study the effects of different components, and to make predictions about behaviour.

    Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may include logical models, as far as logic is taken as a part of mathematics. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed.

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