# An Algebraic Structure For Path Schema (Take 1)

$\newcommand{\abs}{\lvert#1\rvert}$

Note I have written a much improved version of this. See the new post.

This post comes from a longish discussion with Fengyang Wang (@TotalVerb), on the JuliaLang Gitter. Its pretty cool stuff.

In general a path can be described as a a heirachical index. It is defined here independent of the object (filesystem, document etc) being indexed. The precise implementation of the algebric structure differs, depending on the Path types in question, eg Filesystem vs URL, vs XPATH.

This defintion is generally applicable to paths, such as:

• File paths
• URLs
• XPath
• JSON paths
• Apache ZooKeeper Paths
• Swift Paths (Server/Container/Psuedofolder/Object)
• Globs

The defintion whch follows provides all the the expected functionality on paths

## Definition Path Schema:

• $$A$$ a set of absolute paths
• $$R$$ a set of relative paths
• $$A$$ and $$R$$ are disjoint
• $$(R, \cdot)$$ a monoid
• $$\cdot$$ is the called pathjoin operation
• $$(R, \cdot)$$ faithfully acts on $$A$$ and also on $$R$$ from the right
• we denote the identity $$I_R \in R$$ (For filesysystem/URL paths this is .)
• $$p$$ an operator, called the parentdir operation
• $$p:\; A\to A$$
• $$p:\; R\to R$$
• In filesystem/URL paths this is often similar to applying the Relative path .., but with some exceptions.
• For example parentdir(.)=. $$\neq$$ ./.. (See below).
• $$b$$ an operator, called the basename operation
• $$b:\; A\cup R \to R$$
• $$\forall(x) \in A\cup R$$ $$p(x) \cdot b(x) = x$$
• That is to say, pathjoin(parentdir(x), basename(x)) == x
• $$\forall r \in A \cup R \wedge p(r)=r \Leftrightarrow b(r)=I_R$$
• and we call such elements $$r$$ roots
• $$\forall x \in A \cup R, \exists n \in \mathbb{N}: s.t. p^n(x) = p^{n-1}$$ – a root
• $$I_R$$ is the only root in $$R$$
• There exist at least one root in $$A$$
• In unix filesystems, this is /
• In windows, this is each drive name C:,D: etc, and a global // for names pipes and UNC paths
• For URLs this is the domain name.
• The monoid $$(R, \cdot)$$ is generated by the set $${x\in R \mid p(x)=I_R}$$.
• We will call this set of generators $$R^\star$$, or the minimal relative path components.

## Derived Operations:

From this we can define additional operations:

• $$root:\; A \cup R \to A \cup R$$ that finds the root of the path.
• $$root(x)=r$$ for $$r:=\; p^n(x)=p^{n+1}(x)$$ which we know exists for some $$n\in \mathbb(N)$$
• As we know from above the root of an elememnt of $$A$$ is an element of $$A$$
• and the root of an elememnt of $$R$$ is $$I_R$$
• $$parts:\:A \cup R \to (A \cup R)_0^n$$: for $$n\in\mathbb{N}$$ mapping from a path, to a finite sequence of path from the path to the root
• $$parts(x)$$ is given by:
• $$a_0 = b(x)$$
• $$a_{n+1} = p(a_n)$$ if $$p(a_{n+1}) \neq a_{n-1}$$ otherwise not defined.
• if $$x \in A$$ then the final element is a root of $$A$$, and all others in $$R^\star$$
• if $$x \in R$$ then the final element is the root of $$R$$ ($$I_R$$), and all elements are in $$R^\star$$
• $$\sum_{i==n}^{i=0} a_i = x$$ for $$parts(x) - (a_i)^n_0$$
• The series of of $$part(x)$$ over the $$\cdot$$ operation is equal to $$x$$

### Resolving a Path to the object

Finally we have an the operation that turns a absolute path ($$x\in A$$) into a entity, or a set of entities. These operations are less clear, as at this level objects must be considered – the data store (the indexed set) must be accessed to resolve. And issues like synlinks, hardlinks etc start to matter (To use examples POSIX filepaths).

• Call this operation $$e$$ – to evaluate the path
• $$e:\; A \to \subseteq \mathcal{P}(D)$$
• for a $$D$$ the set of indexed objects (filesystem, document etc)
• For $$x_1, x_2 \in A$$, $$x_1=x_2 \Rightarrow eval(x_1) == eval(x_2)$$ that is to say $$eval$$ is a function.
• Depending on implementation, this may be 1 enitity, or many
• One might call Path schema with paths which always eval to one or zero entities MonoPath Schemas
• eg URL, POSIX / NT File System paths
• one might call Path schema with paths that can eval to any nymber of entities MultiPaths Schemas
• eg XPATH or Glob
• On the cardinality of $$e(x)$$
• for $$\abs{e(x)}$$ being the number of entities given by $$e(x)$$ – the cardinality of the set
• $$\abs{e(p(x))} \le \abs{e(x)}$$
• $$\abs{e(x)} > 0 \Rightarrow \abs{e(p(x))} > 0$$
• or its (perhaps more interesting) contra-positive: $$\abs{e(p(x))}=0 \Rightarrow \abs{e(x)}=0$$