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

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