Algebra Map

2020 James B. Wilson

Colorado State University

 

What is Algebra?

Varieties of algebras

Building blocks of algebra

Applying algebra

What is Algebra?

  • Types \(A\) of data equipped with (homogeneous) operators \(A^n\to A\)
  • Theory: Congruence=Quotient=Homomorphism

Varieties of algebras

Building blocks of algebra

Applying algebra

What is Algebra?

Varieties of algebras

  • All algebras presentable with polynomials
  • Theory: classes of algebra closed to products, subalgebras, quotients have polynomial laws.

Building blocks of algebra

Applying algebra

What is Algebra?

Varieties of algebras

Building blocks of algebra

  • Simple algebras have no quotients, degenerate simple drop some operators, full simples use all.
  • Theory: Classify & Sort: full simples on top, degenerate below

Applying algebra

What is Algebra?

Varieties of algebras

Building blocks of algebra

Applying algebra

  • Find simple and solve polynomial equations there
  • Spot obstructions to lifting claims of simples
  • Obstructions become a new algebra and we start over

Algebra in action

How algebra acts.

Varieties of actions

Building blocks of actions

Applying actions

How algebra acts.

  • Many types \(A_1,A_2,\ldots\) and heterogeneous operators \(A_n\times \cdots \times A_1\to A_0\)
  • Theory. Shuffle terms to make operators and homomorphisms=quotients=congruences.

Varieties of actions

Building blocks of actions

Applying actions

How algebra acts.

Varieties of actions

  • Mimic polynomial equations.
  • Theory: every action represented by endomorphisms

Building blocks of actions

Applying actions

How algebra acts.

Varieties of actions

Building blocks of actions

  • Simples have no quotients, some degenerate, some full.
  • Theory: classify & sort the simples

Applying actions

How algebra acts.

Varieties of actions

Building blocks of actions

Applying actions

  • Build algorithms to identify simple actions
  • Turn simple actions into congruences of algebra, now return to the study of the algebra