Homotopy Type Theory Algebra > history (Rev #11)

Here we collect articles about doing set-level algebra in HoTT.

Decidable objects

• booleans
• decidable universal quantifier
• decidable existential quantifier
• decidable directed graph
• decidable setoid
• decidable set
• decidable preordered type
• decidable strict order
• decidable dense strict order?

Inverse images

• inverse image
• iterated inverse image
• infinitely iterated inverse image

Magmas

• monoid
• commutative monoid
• group
• abelian group/Z-module
• divisible group
• halving group/D-module
• torsion-free divisible group/Q-module
• torsion-free halving group
• action
• magma
• power-associative magma
• abelian group homomorphism
• bilinear function

Rings, modules, and algebras

• Z-module/abelian group
• Q-module/torsion-free divisible group
• abelian group homomorphism
• bilinear function
• Z-algebra
• Q-algebra
• square function
• quadratic function
• characteristic
• unital Z-algebra
• ring
• commutative ring
• algebra (ring theory)
• commutative algebra (ring theory)
• polynomial ring
• univariate polynomial ring
• power function
• polynomial function
• function algebra
• module
• graded module
• filtered algebra
• quadratic form
• diagonal
• diagonal form?
• Clifford algebra
• geometric algebra
• algebra (module theory)

Sequences

• sequence

• left shift operator

• right shift operator

• Z-module

• sequential polynomial

• series operator

• inverse series operator

• sequential derivative

• Q-module

• sequential antiderivative

• sequential differential operator?

• sequential linear differential equation?

• sequential exponential?

• sequential sine?

• sequential cosine?

• bidirectional sequence

• bidirectional shift operators?

Ordered commutative rings

• ordered integral domain
• Archimedean ordered integral domain
• ordered field
• Archimedean ordered field
• sequentially Cauchy complete Archimedean ordered field

Cancellation algebras

to be ported to the nlab

• cancellation Z-algebra
• residue cancellation Z-algebra?
• Heyting cancellation Z-algebra
• discrete cancellation Z-algebra

References

• HoTT Book
• Henri Lombardi, Claude Quitté, Commutative algebra: Constructive methods. Finite projective modules, arXiv:1605.04832