# 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
Quotient structures
Extending the language by defined structures
Concrete categories
Other concepts of category
Morphisms between systems with several types
3.2. Special morphisms
Functions defined by composition
Monomorphism, Epimorphism, Section, Retraction, Isomorphism, Endomorphism, Automorphism, Embedding, Elementary embedding
Initial and final objects
3.3. Algebras
Algebra
Morphisms of algebras
Subalgebras
Images of algebras
Preimages of subalgebras.
Intersections of subalgebras.
Subalgebra generated by a subset
Minimal subalgebra.
Injective, surjective algebras
3.4. Algebraic terms and term algebras
Algebraic drafts
Sub-drafts and terms
Categories of drafts
Intepretations of drafts in algebras
Term algebras
3.5. Integers and recursion
The set ℕ
Recursively defined sequences
Multiplication
A more general form of recursion
Commutativity, associativity and commutants
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

Monoids and groups
Abstract monoids
Abstract groups
Submonoids and morphisms of monoids
Actions of monoids and groups
Left action
Typical examples
Right actions
Centralizers
Representation theorem
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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