User:Lord Farin/Long-Term Projects/Howson

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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)
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