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
Relational systems
Concrete categories
Preserved structures
Preservation of some defined structures
Rebuilding structures in a concrete category
Categories of typed systems
3.2. Algebras
Morphisms of algebras
Images of algebras
Stable subsets
Preimages of stable subsets
Intersections of stable subsets
Subalgebra generated by a subset
Minimal subalgebra
Injective, surjective algebras
Injectivity lemma
Schröder–Bernstein theorem
3.3. Special morphisms
Quotient systems
Strong preservation, embeddings and isomorphisms
Embeddings of algebras
Elementary embeddings
Elementary equivalence
Endomorphisms, Automorphisms
The Galois connection (End, Inv)
3.4. Monoids
Transformations monoids
Commutants and centralizers
Other concepts of submonoids and morphisms (anti-morphisms)
3.5. Actions of monoids
Left actions
Right actions
Effectiveness and free elements
Acts as algebraic structures
Trajectories by commutative monoids
3.6. Invertibility and groups
Permutation groups
Special actions
The Galois connection (Aut, sInv)
3.7. Categories
Functions defined by composition
Monomorphism, Epimorphism
Section, Retraction
Representation theorem
3.8. Initial and final objects
Embeddings in concrete categories
Products in concrete categories
Products in categories
Categories of acts
3.9. Eggs, basis, clones and varieties
Basis and algebraic structures
Abstract clones and varieties

4. and 5. are not required to continue with the following texts on algebra
4. Arithmetic and first-order foundations
5. Model Theory

The following texts are drafts, not well classified yet

Products of relational systems
Truth of formulas in products
Morphisms into products
Products of algebras
Polymorphisms and invariants
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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