# User:Lord Farin/Long-Term Projects/Howson

# Processing of 'A Handbook of Terms used in Algebra and Analysis'

First page it covers: Definition:Statement.

A reference book for undergraduate maths.

## Progress thus far

Initial set-up complete. — Lord_Farin (talk) 16:53, 7 April 2014 (UTC)

Up to $\S 2$. Definition:Set. — Lord_Farin (talk) 14:13, 8 April 2014 (UTC)

Up to $\S 3$. Definition:Relation. — Lord_Farin (talk) 18:02, 8 April 2014 (UTC)

Up to $\S 4$. This concludes the review of the part done by Jshflynn. Definition:Set Partition/Definition 2. — Lord_Farin (talk) 20:20, 8 April 2014 (UTC)

Up to $\S 4$: The rational integers. Definition:Integer. — Lord_Farin (talk) 21:16, 9 April 2014 (UTC)

Up to $\S 6$. Definition:Ring with Unity. — Lord_Farin (talk) 08:20, 10 April 2014 (UTC)

Up to $\S 7$. Definition:Homomorphism (Abstract Algebra). — Lord_Farin (talk) 10:59, 10 April 2014 (UTC)

Up to $\S 8$. Definition:Vector Space. — Lord_Farin (talk) 12:21, 10 April 2014 (UTC)

## Missing Proofs

*None atm*

## Skipped thus far (that is, what needs to be done still)

The following notions have not been introduced on $\mathsf{Pr} \infty \mathsf{fWiki}$ yet, and Howson is only a reference work, hence unsuitable as a primary source:

$\S 1$: Some mathematical language

- Variables and quantifiers
- Definition:Closed Formula (no free vars)
- Definition:Open Formula (free vars)
- Definition:Solution Set (for relations with free vars)

- Axiom systems
- Definition:Satisfiable Proof System (one which has a model)
- Definition:Consistent Proof System (one for which $\varnothing \not\vdash \phi$ for some $\phi$)
- Definition:Relative Consistency (consistent embedding of one proof system into another)
- Definition:Independent Axiom (not derivable from others)
- Definition:Independent Proof System (all axioms independent)
- Definition:Categorical Axiom System (all models isomorphic)

$\S 4$: Number systems I

- Peano's Axioms
- Definition:Isotonic (the operation, if a relation is compatible with it)

- A set-theoretic approach
- Cardinality of Power Set of Finite Set for infinite sets
- Definition:Finite Set (corresponding to D-infinite)
- Stuff on (in)finite cardinals
- $|\R| = 2^{\aleph_0}$

- The rational integers
- Definition:Negative to denominate $-x$

$\S 5$: Groups I

- Definition:Periodic Group (all elements finite order)
- Definition:Torsion-Free Group (only $e$ finite order; also called
**aperiodic**)

- Subgroups
- Definition:Extension of Group (converse relation to subgroup)

$\S 6$: Rings and fields

- Product of Ideals is Ideal
- GCD and LCM in PID, for multiple elements at once
- Definition:Prime Element, intricacies with "prime" and "irreducible" to be addressed

$\S 7$: Homomorphisms and quotient algebras

- Definition:Exact Sequence of Group Homomorphisms, $\ker g = \operatorname{im} f$ (also "exact pair")
- Definition:Short Exact Sequence of group homomorphisms

## Other things

*None*