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
Morphisms
Concrete categories
Preserved structures
Rebuilding structures in a concrete category
Preservation of some defined structures
Categories of typed systems
3.2. Algebras
Algebra
Morphisms of algebras
Subalgebras
Images and preimages of subalgebras
Intersections of subalgebras.
Subalgebra generated by a subset
Minimal subalgebra.
Injective, surjective algebras
Injectivity lemma
3.3. Special morphisms
Isomorphism, Endomorphism, Automorphism
Strong preservation, Embedding
Elementary embedding
Elementary equivalence
Quotient systems
3.4. Monoids
Transformations monoids
Permutation groups
Trajectories
Monoids
Cancellativity
Submonoids and morphisms of monoids
3.5. Actions of monoids
Left actions
Effectiveness and free elements
Right actions
Commutants
Centralizers
Representation theorem
3.6. Categories
Representation of small categories
Functions defined by composition
Monomorphism, Epimorphism
Section, Retraction
Initial and final objects
3.7. Algebraic terms and term algebras
Algebraic drafts
Sub-drafts and terms
Categories of drafts
Intepretations of drafts in algebras
Operations defined by terms
Term algebras
Role of term algebras as sets of all terms
The monoid of unary terms
3.8. Integers and recursion
The set ℕ
Recursively defined sequences
Addition
Multiplication
A more general form of recursion
Interpretation of first-order formulas
3.9. Arithmetic with addition
First-order theories of arithmetic
Presburger arithmetic
The order relation
Arithmetic with order

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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