The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook Monad.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

Introduction

This is the second part of a series of posts describing how I will lead up to introducing monads in my proposed e-book Abstracting Algebra (AbAl). It follows Monads for high school I. Comments in red are meta and mostly will not be included in the book.  

Lists 

A list is a specific kind of mathematical object. This is a reasonable specification for lists:

A list of length $n$ determines and is determined by what its first, second, $\ldots$, $n$th entries are. 

In this post, lists will always be finite in length.

For doing rigorous proofs you need a precise definition of a list, such as a function from $\{1,2,…,n\}$ to a set, or a recursive definition.  This book is not about proofs.

Terminology and representation

The most common way in the symbolic language of math to represent a finite list is to use a comma-delimited expression in parentheses.  For example, \[(4,4,2,8)\] is the list of length 4 whose first and second entries are both $4$, third entry $2$ and fourth entry $8$.

• The order matters and repetitions are allowed. For example, $(4,4,2,8)$, $(4,2,8)$ and $(4,2,4,8)$ are all different lists.
• Other words for lists are (finite) sequence, word, tuple and string.
• Many mathematicians would call $(4,4,2,8)$ an $4$–tuple.
• My Discrete math classnotes discusses the specification and the definition of lists called tuples there) at length on pages 50ff. This section of AbAl will incorporate some of the information there.
• Some computer languages represent our list without the commas: $(4\,\,4\,\,2\,\,8)$.
• Mathematica represents it this way: $\{4,4,2,8\}$.  This conflicts with the usual set notation, where the order does not matter and where repetitions are ignored  — the set $\{4,4,2,8\}$ has three elements.  But if you type Length[$\{4,4,2,8\}$] in Mathematica, you get the answer 4.
• A list of characters (alphabetical, numerical, or other symbols) can be represented  by writing the characters down in order without spaces between them.  For example $(a,a,c,d)$ would be written "aacd".  This representation is referred to as a string or as a word in computing science.  The string "4428" is the base-10 representation of the integer $4,428$.  Of course, it is also the hexadecimal representation of the integer $17,448$. 
• In the text, I will mostly use a cartouche representation: for example, $\boxed{1\ 2\ 3\ 4}$ is the list consisting of the first four positive integers in order.
• The cartouche is more in-your-face than the other representations I've listed and as far as I know is not used to mean anything else.  I'm not sure I can give any better explanation for why I prefer it than that.  Math is supposed to be explicit and precisely defined and justified by clear reasoning, but after all deciding which representation to use is not math, it is art.

Lists with entries from a given set

If $S$ is any set, finite or infinite, $\textrm{Lists}(S)$ denotes the set of all lists of finite length whose entries come from $S$.  Thus the set $\textrm{Lists}(\{1,\  2,\  3\})$ contains:

• $\boxed{2\ 2\ 3\ 2\ 2\ 1}$,
• $\boxed{3\  3\  3\  3}$,
• the list of length $42$ whose first entry is $3$ and every other entry is $1$,
• the empty list $\boxed{\vphantom{n}}$,
• the singleton lists $\boxed{1}$,  $\boxed{2}$ and  $\boxed{3}$, and
• an infinite number of other lists, 
• but the list $\boxed{4\  2\  3}$ is not an element of $\textrm{Lists}(\{1,\  2,\  3\})$.

$\textrm{Lists}$ is a function from sets to sets.  Its input is any set and its output is the set of all finitely-long lists whose entries are from the input set. We will also use the similar function $\textrm{Lists}^+$ which takes a set to the set of nonempty lists with entries from the set.

Associativity

(Review from Monads for high school I.)  If a binary operation is associative, then the operation is defined on any (finite) list of inputs in its underlying set.  For example, the sum of the list $\boxed{4\ 4\ 2\ 8}$  is 18.  It follows from associativity that you can add it up as $(4+4)+(2+8)$, $4+(4+(2+8))$, $4+((4+2)+8)$, $(4+(4+2))+8$ or as $((4+4)+2)+8$.  They all give the same answer. In other words, Plus is in fact an operation on lists of numbers.  It is customary to extend associative binary operations to lists of length $0$ and $1$ by setting the value at the empty list to be the identity element of the operation, and the value at a one element list to be its only entry.  Thus Plus($\boxed{4\ 4\ 2\ 8}$)$=18$, Plus($\boxed{\ \vphantom{0} }$)$=0$, Times($\boxed{\ \vphantom{0} }$)$=1$ and Plus($\boxed{3 }$)$=3$.

Operations defined on finite lists

 You can join two lists together in order to make one list.  

The order matters.  If you join $\boxed{5\ 7}$ to $\boxed{2\ 12\ 7 }$ in that order you get $\boxed{5\ 7\ 2\ 12\ 7}$.  

Join is in fact an associative binary operation on lists.  Example: 

This means we can define an operation on lists of lists that joins all the lists inside together to make one list. 

 Notice the blue rectangle disappears when you do the operation. What I have defined here is a function that has a list of lists as input and a list of numbers as output.

The operation of joining lists to get a single list has a property shown by the drawing below (which will be interactive when I work on it some more).  Start on the left with a list of lists of lists.  The border colors distinguish the innermost lists, bordered in black, from the second level lists, in blue, and the outside list, bordered in green.

• There is only one outside list: It is a list of (blue) lists.  That is the kind of list you can apply join to, so when you do you get a single blue list with five lists inside it (on the bottom of the diagram). "Join outside first" means "apply join to the outside list first". 
• The single blue list on the bottom is again the kind of list you can apply join to, and when you do you get the lower list on the right end of the diagram.
• However, the green list also contains two lists each of which is a list of lists that you can apply join to.  Apply it to both of them and you get the list at the top of the diagram.  
• Again, that list is the kind you can apply join to and when you do you get the upper list on the right.

The two lists on the right are the same.  That always happens, whatever lists you start with.  (Try it with others, and include some singleton and empty lists while you are at it.) 

You might not have thought of this property, and now that you see it, it may look like some sort of second-rate phenom to take note of.  Or not.  But in fact, it turns out that it means that our modest function  $\textrm{Lists}^+$, that takes a set to the nonempty set of lists of its elements is a monad.  (So is $\textrm{Lists}$.) In order to say this we must define some other concepts: functor and natural transformation, and we have to verify a number of other properties of the $\textrm{List}^+$ function:  It is not just a function, it is a functor on the category of sets, the join function is a natural transformation, and some other technicalities.

Once we do that, we can define what the algebras of the join monad are, and it turns out that they are exactly all the associative binary operations.  

In other words:

The binary operation of join on nonempty lists is the mother of all associative binary operations.

But that will have to wait for the next post.

References

• Associative in Wikipedia.
• Binary operation in Wikipedia.
• Discrete math classnotes, by Charles Wells.
• Toposes, triples and theories, by Michael Barr and Charles Wells.