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

Rebuilding structures in a concrete category

Preservation of some defined structures

Categories of typed systems

Algebra3.3. Special morphisms

Morphisms of algebras

Subalgebras

Images and preimages of subalgebras

Intersections of subalgebras.

Subalgebra generated by a subset

Minimal subalgebra.

Injective, surjective algebras

Injectivity lemma

Isomorphism, Endomorphism, Automorphism3.4. Monoids

Strong preservation, Embedding

Elementary embedding

Elementary equivalence

Quotient systems

Transformations monoids3.5. Actions of monoids

Permutation groups

Trajectories

Monoids

Cancellativity

Submonoids and morphisms of monoids

Left actions3.6. Categories

Effectiveness and free elements

Right actions

Commutants

Centralizers

Representation theorem

Representation of small categories3.7. Algebraic terms and term algebras

Functions defined by composition

Monomorphism, Epimorphism

Section, Retraction

Initial and final objects

Algebraic drafts3.8. Integers and recursion

Sub-drafts and terms

Categories of drafts

Intepretations of drafts in algebras

Operations defined by terms

Term algebras

Role of term algebras as sets of all terms

The monoid of unary terms

The set ℕ3.9. Arithmetic with addition

Recursively defined sequences

Addition

Multiplication

A more general form of recursion

Interpretation of first-order formulas

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

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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