Algebra is a field of mathematics which is not
rigorously delimited but can roughly be described as a focus, inside set theory, on
a range of remarkable concepts and tools providing preliminary tools for model
theory (the study of theories and
their models).
Here is a series of texts on the foundations of
algebra, that will be progressively developed.
Set theory and Foundations of Mathematics |
|
| 1.
First
foundations of mathematics 2. Set theory |
|
| 3. Algebra 1 (all in 1 file) | |
| 3.1. Galois connections (basic properties ; see text Galois connections for more developments) |
Fixed points, idempotent functions Monotone, antitone, strictly monotone functions Order between functions, extensive functions Galois connections : definition and examples Properties of Galois connections Characterization of closures |
| 3.2. Morphisms
of relational systems and concrete categories |
Languages Relational systems Morphisms Concrete categories Preservation of some defined structures Categories of typed systems |
| 3.3. Algebras | Algebras - Morphisms of algebras
Algebras as relational systems Qualities of systems with algebraic languages Stable subsets and subalgebras Diverse properties of stability Minimal systems Cantor-Bernstein theorem |
| 3.4. Special morphisms | Isomorphisms, endomorphisms, automorphisms Embeddings Elementary embeddings, elementary equivalence The Galois connection (End, Inv) Morphisms of algebras Images and preimages with algebraic languages |
| 3.5. Monoids | Transformations monoids
Monoids Cancellativity Commutants and centralizers Other concepts of submonoids and morphisms Categories |
| 3.6. Actions of monoids and categories |
Actions of monoids Acts as algebraic structures Effectiveness and free elements Trajectories Right actions Actions of categories Some constructions of actions Trajectories of tuples in concrete categories |
| 3.7. Invertibility and groups | Permutation groups Inverses Transpositions Groups Special actions The Galois connection (Aut, sInv) Invertibility of morphisms |
| 3.8. Properties in categories | Monomorphisms, Epimorphisms Sections, Retractions Modules Examples of modules Subobjects Representation theorem |
| 3.9. Initial and final objects | Initial and final objects Eggs Subobjects as sub-co-actions Embeddings in concrete categories Dependencies between some properties of morphisms Equalizers Submodules |
| 3.10. Products of systems | Products of actions Products in categories Products in concrete categories Products of modules Products of relational systems Products of algebras Fiber products Intersections of subobjects Intersections of submodules Intersections of subsystems |
| 3.11. Basis |
Eggs as acting monoids Algebraic structures on modules Basis Coproducts Basis and coproducts Equational systems |
| 3.12. Composition of relations | The category of relations Re-expressing properties of relations Generated preorders Generated stable subsets Transitive closure Well-founded relations Well-order Greatest and least elements The successor function |
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4. Arithmetic and first-order foundations 5. Second-order foundations |
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4. and 5. are not required to continue with the following texts on algebra
The Galois connection Inv-Pol between sets of operations and relationsDuality systems and theories
The Galois connection Pol-Pol between sets of operations
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and foundations of mathematics