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. &mdash; Lord_Farin (talk) 16:53, 7 April 2014 (UTC)

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

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

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

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

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

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

Up to $\S 8$. Definition:Vector Space. &mdash; 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 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 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