# 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

Addition

Multiplication

A more general form of recursion

Commutativity, associativity and commutants

3.6. Arithmetic with addition

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

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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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