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.

Languages, systems3.2. Algebras

Morphisms

Extending the language by defined structures

Concrete categories

Other concepts of category

Morphisms between systems with several types

Algebra3.3. Special morphisms

Morphisms of algebras

Subalgebras

Images of algebras

Preimages of subalgebras.

Intersections of subalgebras.

Subalgebra generated by a subset

Minimal subalgebra.

Injective, surjective algebras

Functions defined by composition3.4. Monoids and actions

Monomorphism, Epimorphism, Section, Retraction, Isomorphism, Endomorphism, Automorphism, Embedding, Elementary embedding

Quotient systems

Initial and final objects

Transformations monoids3.5. Categories

Monoids

Submonoids and morphisms of monoids

Left and Right actions

General examples

Centralizers

Representation theorem

Small categories and representation3.6. Algebraic terms and term algebras

Functions defined by composition

Monomorphism, Epimorphism

Initial and final objects

Algebraic drafts3.7. Integers and recursion

Sub-drafts and terms

Categories of drafts

Intepretations of drafts in algebras

Term algebras

The set ℕ3.8. Arithmetic with addition

Recursively defined sequences

Addition

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.

Groups, automorphisms and invariants

Abstract groupsProducts of relational systems (updated on Sept. 2014)

...Actions of groups...

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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