Foundations of Algebra

Algebra is a field of mathematics which is not rigorously delimited but can roughly be described as a focus on a range of some remarkable concepts and tools concerning model theory (the study of theories and their models in the context of set theory).

Here is a series of texts on the foundations of algebra, that will be progressively developed.

3. Algebra 1

3.1. Morphisms of relational systems and concrete categories
Languages, systems
Extending the language by defined structures
Concrete categories
Other concepts of category
Morphisms between systems with several types
3.2. Algebras
Morphisms of algebras
Images of algebras
Preimages of subalgebras.
Intersections of subalgebras.
Subalgebra generated by a subset
Minimal subalgebra.
Injective, surjective algebras
3.3. Special morphisms
Functions defined by composition
Monomorphism, Epimorphism, Section, Retraction, Isomorphism, Endomorphism, Automorphism, Embedding, Elementary embedding
Quotient systems
Initial and final objects
3.4. Monoids and actions
Transformations monoids
Submonoids and morphisms of monoids
Left and Right actions
General examples
Representation theorem
3.5. Categories
Small categories and representation
Functions defined by composition
Monomorphism, Epimorphism
Initial and final objects
3.6. Algebraic terms and term algebras
Algebraic drafts
Sub-drafts and terms
Categories of drafts
Intepretations of drafts in algebras
Term algebras
3.7. Integers and recursion
The set ℕ
Recursively defined sequences
A more general form of recursion
Commutativity, associativity and commutants
3.8. Arithmetic with addition
First-order arithmetic
Presburger arithmetic
The order relation

Next comes 4. Model Theory, which is not required to continue with the following texts on algebra.

The following texts are drafts, not well classified yet

The Galois connection (Aut, sInv) between structures and permutations

Groups, automorphisms and invariants
Abstract groups
...Actions of groups...
Products of relational systems (updated on Sept. 2014)
Truth of formulas in products
Morphisms into products
Products of algebras
Polymorphisms and invariants (updated on Sept. 2014)
The Galois connection Inv-Pol between sets of operations and relations
The Galois connection Pol-Pol between sets of operations
Duality systems and theories

Vector spaces and their dualities

References of works by other authors on duality systems:
Sir-algebras (with also a list of Galois connections on page 2)
On duality theories


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