# Foundations of Algebra

Algebra is a field of mathematics which is not rigorously delimited but can roughly be described as a focus, inside set theory, on a range of remarkable concepts and tools providing preliminary tools for model theory (the study of theories and their models).
Here is a series of texts on the foundations of algebra, that will be progressively developed.

### Set theory and Foundations of Mathematics

1. First foundations of mathematics
2. Set theory
3. Algebra 1 (all in 1 file)
3.1. Galois connections

(basic properties ; see text Galois connections
for more developments)
Fixed points, idempotent functions
Monotone, antitone, strictly monotone functions
Order between functions, extensive functions
Galois connections : definition and examples
Properties of Galois connections
Characterization of closures
3.2. Morphisms of relational systems
and concrete categories
Languages
Relational systems
Morphisms
Concrete categories
Preservation of some defined structures
Categories of typed systems
3.3. AlgebrasAlgebras - Morphisms of algebras
Algebras as relational systems
Qualities of systems with algebraic languages
Stable subsets and subalgebras
Diverse properties of stability
Minimal systems
Cantor-Bernstein theorem
3.4. Special morphisms Isomorphisms, endomorphisms, automorphisms
Embeddings
Elementary embeddings, elementary equivalence
The Galois connection (End, Inv)
Morphisms of algebras
Images and preimages with algebraic languages
3.5. MonoidsTransformations monoids
Monoids
Cancellativity
Commutants and centralizers
Other concepts of submonoids and morphisms
Categories
3.6. Actions of monoids and categories Actions of monoids
Acts as algebraic structures
Effectiveness and free elements
Trajectories
Right actions
Actions of categories
Some constructions of actions
Trajectories of tuples in concrete categories
3.7. Invertibility and groupsPermutation groups
Inverses
Transpositions
Groups
Special actions
The Galois connection (Aut, sInv)
Invertibility of morphisms
3.8. Properties in categories Monomorphisms, Epimorphisms
Sections, Retractions
Modules
Examples of modules
Subobjects
Representation theorem
3.9. Initial and final objects Initial and final objects
Eggs
Subobjects as sub-co-actions
Embeddings in concrete categories
Dependencies between some properties of morphisms
Equalizers
Submodules
3.10. Products of systems Products of actions
Products in categories
Products in concrete categories
Products of modules
Products of relational systems
Products of algebras
Fiber products
Intersections of subobjects
Intersections of submodules
Intersections of subsystems
3.11. Basis Eggs as acting monoids
Algebraic structures on modules
Basis
Coproducts
Basis and coproducts
Equational systems
3.12. Composition of relationsThe category of relations
Re-expressing properties of relations
Generated preorders
Generated stable subsets
Transitive closure
Well-founded relations
Well-order
Greatest and least elements
The successor function
4. Arithmetic and first-order foundations
5. Second-order foundations

4. and 5. are not required to continue with the following texts on algebra

### The following texts are drafts, not well classified yet

Relational clones
Truth of formulas in products
Abstract clones
Clones
Abstract clones and varieties
Correspondences between interpretations of formulas
Centralizers in abstract clones
The multiplication monoid in an abstract clone
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

### References of works by other authors

Notes on semigroups by Uday S. Reddy
A Theory of Transformation Monoids: Combinatorics and Representation Theory by Benjamin Steinberg

On duality systems: