Abstract algebra

This is Abstract Algebra, an advanced set of topics related to Algebra. Readers of this book are expected to have read and understand the information presented in the Algebra, and Linear Algebra books.

Abstract algebraic systems

An Abstract Algebraic System is a tuple of the form $S=(\underset{\_}{S},f_1,...,f_n,R_1,...,R_k,\nu )$. In this $\underset{\_}{S}$ is a set, $f_i$s are function symbols of some arity , $R_i$s are predicate symbols of some arity and $\nu$ is an interpretation of these symbols on the set.

If no predicate symbols are present then $S$ is an Algebra (or a Universal Algebra). We allow 0-ary operations in the above, so distinguished elements must be interpreted as 0-ary operations.

• /Equivalence relations and congruence classes/

Group theory

• /Sets and Compositions/
• /Groups/

Rings

• /Rings, fields and modules/
• /Ring Homomorphisms/
• /Ideals/
• /Modules/
• Miscellaneous Topics
  • /Integral domains/

Fields

• /Fields/
• /Factorization/

Vector Spaces

• /Vector Spaces/

Algebras

• /Algebras/
  • /Boolean algebra/
  • /Clifford Algebras/
  • /Quaternions/

Modules

• /Modules/
• /Hypercomplex numbers/

Further abstract algebra

• /Category theory/
• /Lattice theory/
• /Matroids/

Authors

• A. Mani
• Jaden Mathos
• Richard Campanha