erdbeere

ErDBeere is short for “Erkenntnisfördernde Datenbank zur Beispielerfassung und -entwicklung” (informative database for the development and management of examples) and should be thought of as an e-learning tool for higher (i. e., University level) mathematics.

It aims to store mathematical examples somewhat like the ring database and is supposed to help students explore features of mathematical objects. Consider the following questions:

• What does a principal ideal domain that is not Euclidean look like?
• What does a quasi-projective and proper morphism between schemes look like, that isn't projective?
• Is there a ring that is a principal ideal domain, but not a unique factorization domain?

The first question has a simple enough answer with the ring of integers of Q(√-19). The second also has an example as the answer, but it is worthwhile to know that every quasi-projective proper morphism X→Y is projective if Y is qcqs. The third question of course has no example at all, as every principal ideal domain is a UFD.

ErDBeere wants to represent all the data in the answers above, i.e., concrete examples and abstract implications, in the nicest possible manner.

Installation

Installation is easiest via docker (although git clone && bundle install should also work just fine.) Use the environment variable SECRET_KEY_BASE.

The following commands should yield a working installation:

docker create -e "SECRET_KEY_BASE=verylongsecret" --name erdbeere -p 3000:3000 oqpvc/erdbeere
docker start erdbeere

Data Structures

The “nicest possible manner” is of course a Web 2.0 application. This is hence a Ruby on Rails project, which isn't all that suitable to represent the aforementioned data — especially the mathematical implications.

We will explain the internal data structures using example pieces of code.

Categories, Properties and Implications

ring = Structure.create do |s|
  s.name = 'Ring'
  s.definition = 'A ring $R$ is an abelian group together with a ' +
                 'map $R\times R …' 
end

unitary = Property.create do |p|
  p.name = 'unitary'
  p.structure = ring
  p.definition = '…'
end

# …

l_noeth = Property.create(name: 'left Noetherian', structure: ring)
r_noeth = Property.create(name: 'right Noetherian', structure: ring)
comm = Property.create(name: 'commutative', structure: ring)
vnr = Property.create(name: 'von Neumann regular (aka absolutely flat)',
                        structure: ring)

Rings are easy classes of objects to represent, as they don't depend on other structures. But as soon as $R$-modules enter the picture, this becomes decidedly more complicated, as their properties may depend on their ground ring. Structures can hence have building blocks.

rmod = Structure.create(name: '$R$-(left-)Module')
base_ring = BuildingBlock.create(name: 'base ring',
                                   explained_structure: rmod,
                                   structure: ring, definition: 'A ' +
                                   'ring homomorphism $R\longrightarrow ' +
                                   '\mathrm{End}(M)$ …')

Now we want to encode the following result: “A finitely generated $R$-left-module over a left Noetherian ring is itself Noetherian”.

fin_gen = Property.create(name: 'finitely generated', structure: rmod)
noeth_module = Property.create(name: 'ascending chain condition for ' +
                               'submodules', structure: rmod)

base_ring_is_lnoeth = Atom.create(stuff_w_props: base_ring, satisfies:
                                   l_noeth)
module_is_fg = Atom.create(stuff_w_props: rmod, satisfies: fin_gen)
module_has_acc = Atom.create(stuff_w_props: rmod, satisfies: noeth_module)

[module_is_fg, base_ring_is_lnoeth].implies! module_has_acc

Examples

An example for a structure in the sense above depends on examples of all of its building blocks (e. g., for an $R$-module we first need a ring $R$). Consider the following code snippet:

zee = Example.create(structure: ring)
zee.satisfies! [comm, l_noeth]
zee.violates! vnr

zee_r = Example.create(structure: rmod)
BuildingBlockRealization.create(example: zee_r, building_block:
  base_ring, realization: zee)
zee_r.satisfies! module_is_fg

Using the logic engine

The internal logic engine is as stupid as one get away with. I assume that it is hence very slow, which still might be fast enough for practical purposes. Working with it roughly looks like this:

[comm.to_atom, l_noeth.to_atom].is_equivalent! [comm.to_atom, r_noeth.to_atom]

zee_r.satisfies?(module_has_acc)
zee_r.satisfies?(Atom.find_or_create_by(stuff_w_props:
                                                   base_ring,
                                                   satisfies:
                                                   r_noeth))
zee_r.satisfied_atoms.each do |a|
  puts a.to_s
end

As one can see, the implications on the level of examples work recursively: Even though we never had an ExampleFact that stated that for commutative base rings (and not rings) left and right Noetherian are equivalent, the logic engine derives those kind of statements from the bottom up.