Mathematical Structure

A Mathematical Structure is a mathematical object consisting of a set together with operations and relations that provides semantic interpretation (designed to serve as a model for formal mathematical systems and enable concrete mathematical reasoning).

• AKA: Mathematical Model, Algebraic Structure.
• Context:
  • It can typically satisfy Mathematical Axioms through mathematical property instantiation.
  • It can typically provide Semantic Interpretations for formal mathematical symbols.
  • It can typically exhibit Mathematical Properties through mathematical relational configurations.
  • It can typically support Mathematical Operations through mathematical function definitions.
  • It can typically enable Truth Evaluation for mathematical statements.
  • It can typically serve as Foundation Elements for mathematical metamodels.
  • It can typically define Mathematical Universes for mathematical model instantiation.
  • ...
  • It can often model multiple Formal Mathematical Systems through mathematical interpretation mappings.
  • It can often possess Invariant Properties under mathematical structure-preserving transformations.
  • It can often participate in Category-Theoretic Relationships with mathematical morphisms.
  • It can often admit Automorphism Groups preserving mathematical structural features.
  • It can often provide Semantic Domains for mathematical metamodel interpretation.
  • It can often establish Mathematical Constraints for mathematical model families.
  • ...
  • It can range from being a Finite Mathematical Structure to being an Infinite Mathematical Structure, depending on its mathematical carrier set cardinality.
  • It can range from being a Simple Mathematical Structure to being a Complex Mathematical Structure, depending on its mathematical operation and relation count.
  • It can range from being a Discrete Mathematical Structure to being a Continuous Mathematical Structure, depending on its mathematical topological properties.
  • It can range from being a First-Order Mathematical Structure to being a Higher-Order Mathematical Structure, depending on its mathematical logical complexity.
  • ...
  • It can serve as Model Instances for formal mathematical systems.
  • It can be analyzed through Mathematical Structure Theory for mathematical classification purposes.
  • It can be compared via Mathematical Homomorphisms and mathematical isomorphisms.
  • It can underpin Mathematical Metamodel Frameworks through mathematical semantic grounding.
  • ...
• Examples:
  • Algebraic Mathematical Structures, such as:
    • Group Mathematical Structures, such as:
      • Integers Under Addition, with mathematical identity element zero.
      • Symmetric Group, with mathematical permutation composition.
      • Matrix Group, with mathematical matrix multiplication.
    • Ring Mathematical Structures, such as:
      • Integer Ring, with mathematical addition and multiplication operations.
      • Polynomial Ring, over mathematical coefficient fields.
      • Matrix Ring, with mathematical matrix operations.
    • Field Mathematical Structures, such as:
      • Real Number Field, with complete ordered mathematical field property.
      • Complex Number Field, with mathematical algebraic closure.
      • Finite Field, with mathematical prime power order.
  • Order Mathematical Structures, such as:
    • Partially Ordered Sets, such as:
      • Power Set with Inclusion, ordered by mathematical subset relation.
      • Natural Numbers with Divisibility, ordered by mathematical divisibility relation.
      • Function Space with Pointwise Order, ordered by mathematical function comparison.
    • Lattice Mathematical Structures, such as:
      • Boolean Algebra, with mathematical complement operation.
      • Subgroup Lattice, of mathematical group structures.
      • Concept Lattice, from mathematical formal contexts.
  • Topological Mathematical Structures, such as:
    • Metric Spaces, such as:
      • Euclidean Space, with standard mathematical distance function.
      • Discrete Metric Space, with mathematical discrete topology.
      • Function Space with Sup Norm, with mathematical uniform convergence.
    • Manifold Mathematical Structures, such as:
      • Differentiable Manifold, with mathematical smooth structure.
      • Riemannian Manifold, with mathematical metric tensor.
      • Complex Manifold, with mathematical holomorphic structure.
  • Relational Mathematical Structures, such as:
    • Graph Mathematical Structures, with mathematical vertex and edge relations.
    • Database Relations, with mathematical tuple structures.
    • Equivalence Relations, with mathematical partition structures.
  • Model-Theoretic Mathematical Structures, such as:
    • First-Order Structures, interpreting mathematical predicate logic.
    • Multi-Sorted Structures, with mathematical type systems.
  • ...
• Counter-Examples:
  • Formal Mathematical System, which provides syntactic framework rather than mathematical semantic content.
  • Mathematical Proof, which demonstrates logical derivation rather than mathematical structural property.
  • Mathematical Algorithm, which specifies computational procedure rather than mathematical static structure.
  • Mathematical Notation, which provides symbolic representation rather than mathematical semantic structure.
• See: Formal Mathematical System, Model Theory, Universal Algebra, Category Theory, Structure-Preserving Map, Mathematical Metamodel.

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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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