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.

Languages3.2. Algebras

systems

Morphisms

Concrete categories

Preserved structures

Preservation of some defined structures

Rebuilding structures in a concrete category

Categories of typed systems

Algebra3.3. Special morphisms

Morphisms of algebras

Subalgebras

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

Quotient systems3.4. Monoids

Strong preservation, embeddings and isomorphisms

Embeddings of algebras

Elementary embedding

Elementary equivalence

Endomorphism, Automorphism

Transformations monoids3.5. Actions of monoids

Trajectories

Monoids

Cancellativity

Commutants and centralizers

Other concepts of submonoids and morphisms

Anti-morphisms

Left actions3.6. Invertibility and groups

Effectiveness and free elements

Acts as algebraic structures

Right actions

Representation theorem

Trajectories by commutative monoids

Permutation groups3.7. Categories

Inverses

Groups

Representation of small categories3.8. Algebraic terms and term algebras

Functions defined by composition

Monomorphism, Epimorphism

Section, Retraction

Initial and final objects

Categories of acts

Algebraic drafts3.9. Integers and recursion

Sub-drafts and terms

Categories of drafts

Intepretations of drafts in algebras

Operations defined by terms

Condensed drafts

Term algebras

Role of term algebras as sets of all terms

The monoid of unary terms

free monoid

The set ℕ3.10. Arithmetic with addition

Recursively defined sequences

Addition

Multiplication

Inversed recursion and relative integers

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.

Groups, automorphisms and invariants

...Products of relational systems (updated on Sept. 2014)

Truth of formulas in productsPolymorphisms and invariants (updated on Sept. 2014)

Morphisms into products

Products of algebras

The Galois connection Inv-Pol between sets of operations and relationsDuality systems and theories

The Galois connection Pol-Pol between sets of operations

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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