In this article we introduce the little known Scandinavian field of Inverterbar Komplekse Entydig Analyse Matematikk1, henceforth IKEA Matematikk. We shall give an overview of the main ideas and results with an outline of the strategies used in their proof.
1Ja visst! Men jeg kan litt norsk og ingen svensk. Også, "Komplekse" staves med "C" på svensk.
The field of Ikea Matematikk is well-known in Scandinavia but seems to be relatively unknown outside. It has produced many surprising results that appear to contradict intuition based on traditional Mathematics such as Euclidean geometry and Archimedean arithmetic.
In this article we shall introduce the main ideas and sketch some of the key developments of the theory. The applications of this area are more well-known than the theoretical which is why we focus primarily on the abstract side of the subject.
2 Complexes and Models
The starting point of IKEA Matematikk is the IKEA complex, which is a very simple concept.
An IKEA complex is a pair of sets . The elements of are called bolts and of are pieces. If both and are finite, we say that the complex is a finite IKEA complex.
A morphism of IKEA complexes is a pair of morphisms of sets, and .
The category of IKEA complexes is thus . The subtleties enter with the notion of instructions for an IKEA complex.
Instructions for an IKEA complex consists of:
A pointed set of tools, ,
A total ordering on the set which describes the order in which the bolts are to be added to the model,
A function which assigns to each bolt a tool,
A pair of functions such that for each , which describes which two pieces a bolt connects.
An IKEA complex together with instructions for that complex is called an IKEA model.
A morphism of IKEA models is a triple of morphisms of sets , , and (basepoint preserving) which commute with the structure.
It's important to note that the obvious functor from IKEA models to IKEA complexes is faithful but not full. This is because of the ordering on the set of bolts which is not part of the data of the complex.
In the early days of the theory there was considerable discussion as to whether the set of tools should have any particular structure. The eventual settlement on a pointed set is in no small part due to Alum's theorem (see Corollary 7 below).
Once we have instructions we can consider the geometric realisation of the complex.
A geometric realisation of an IKEA model , written , is an immersion , where is the unit square, such that:
is affine on each component (so the image of each component is a rectangle),
is an embedding when restricted to the coproduct of the interiors of the squares (so the rectangles can only overlap on their boundaries),
for , the image of under is connected (so when a bolt connects two pieces of wood then then those two pieces are connected toplogically).
There is a generalisation of this definition wherein more shapes than squares are allowed for the models. Let be the set of polygons in .
An uvanlig geometric realisation of an IKEA model consists of a function and an immersion satisfying the same conditions as for a geometric realisation.
The core concept of IKEA Matematikk is the following.
Let be a finite IKEA model, with set of bolts . Let be the size of and for let be the corresponding initial set of (thus and ). Let be the IKEA model formed from by replacing by and the various functions on by their restrictions to .
A construction of the IKEA model consists of a sequence of geometric realisations of such that:
for each , there is a homotopy of geometric realisations ,
the realisation is an embedding on the whole of .
An IKEA model is constructible if it admits a construction.
It is probably worth pointing out that the second condition means that the realisation embeds each copy of into in such a way that the images are disjoint. Without that condition, one could take a geometric realisation of and simply restrict it to each in turn. With that condition, the effectively interpolate between the disjoint embedding of and a full realisation of ; each homotopy involves attaching a piece onto the existent shape.
With this we can state the first theorem of IKEA Matematikk:
Let be an IKEA model with set of tools . Let be another set of tools and a surjection (hereafter called a forgetful function). Define to be with the tool function replaced by its composition with .
If is constructible then so is .
Loosely speaking, this theorem states that if a tool is forgotten then it is still possible to construct the model, albeit with a different realisation. The key to the proof of this theorem is the introduction of an auxiliary component which Bestå referred to as the Banneord set. The forgetting of a tool is offset by an increase in this set.
A simple application of this theorem is the following, although this result predates the above theorem.
If is a finite IKEA model then there is a model derived from with tool set consisting of a single element such that if is constructible then so is .
The original proof followed the standard method of proof by exhaustion.
A singleton tool set is known as a key, in the literature this is sometimes known as an Alum key in honour of the original author.
This has led to the basepoint of a set of tools being referred to as an Alum key. The initial object in the category of IKEA models is therefore ; that is, the initial IKEA model is nothing but an Alum key.
There are many open questions regarding constructibility of IKEA models.
Give an algorithm that produces a construction of a given IKEA model.
It is conjectured that such an algorithm would be NP-complete, but as a general algorithm has not been devised this is unknown.
Much recent work has been on the topic of deconstructibility. Determining the correct notion of a deconstruction of an IKEA model is still open. There are several proposals that are demonstrably not equivalent, ranging from a stepwise reversal of a construction through to the øks method. However, all agree that as with a construction, a deconstruction should involve a series of steps. Given a method of deconstruction, there are several open questions.
For a given definition of deconstructibility:
Can every constructible IKEA model be deconstructed?
For a given IKEA model that can be deconstructed, can it be reconstructed? (An IKEA model that can be deconstructed and then reconstructed is known as inverterbar, whence the inclusion of that word in the name of this field.)
Is there a step beyond which a deconstruction can no longer be reconstructed?
It is known that for the øks method, every IKEA model can be deconstructed but that none can be reconstructed.
All of the above has focussed on finite IKEA models, leading to the obvious area for future research.
What are valid extensions of the theories of constructibility and deconstructibility to infinite IKEA models?
Is there a suitable notion of a semi-infinite IKEA model?
Sadly, the method of proof by exhaustion – which has proven to be so effective in this field – does not apply in the infinite case. Proof by infinite descent is looking more promising.
4 Connections with Other Areas
The field of IKEA Matematikk has many connections with other, more established, areas of Mathematics. Here we shall outline just a few.
A IKEA neighbourhood of an IKEA model is an open subset with the property that there is a construction of the model such that the geometric realisations and homotopies factor through .
The obvious open problem in IKEA Topology is therefore the following.
Given an IKEA model, how do the IKEA neighbourhoods behave? Is the intersection of two IKEA neighbourhoods an IKEA neighbourhood?
How do IKEA neighbourhoods relate to –neighbourhoods of the final construction?
There are obviously many connections with combinatorics.
Given an IKEA model, how many different sets of inequivalent instructions exist for that model?
How many constructible instructions exist?
We have already touched on the connections to algorithm design but there are plenty more. The following is the problem that has motivated the current author in studying this area.
Given a construction of an IKEA model, , and a neighbourhood, , of the final stage, is there an equivalent construction such that is an IKEA neighbourhood for this second construction?
In other words, given a set of instructions that construct an IKEA model, is it possible to modify those instructions so that the model can be built in a confined space that nevertheless should fit the final model?
This question has taxed many workers in this field, and has proved a source of much fruitful thought. Many particular cases have been solved, but the general case is still open. It is to be hoped that by broadening the awareness of this area of Mathematics, more effort can be brought to bear on this most vexing of problems.
It is hoped that this short article has given a brief introduction to the field of IKEA Matematikk. It is an area that admits to much experimentation: it is easy to build simple IKEA Models that can be used to form and test conjectures so there is much to be investigated even by newcomers to the subject. Any of these simple results could potentially benefit the many practitioners so we would like to encourage other Mathematicians to bring their skills to bear on this most rewarding, stylish, and useful field of endeavour.