Frenemy isomorphisms

I was recently introduced by some friends to the concept of the frenemy, a term which has recently even entered public political discourse. I was excited to learn about this new class of social relationship and feel this may revolutionize how I classify my relationships.

However, I wondered whether this concept has yet been fully described. Are its relationships to other categories well-defined, and is it robust enough to undergo the same usages and manipulations as its constituents, ‘friend’ and ‘enemy’? For example, it immediately occurred to me to ask, “Is the frenemy of my frenemy my frenemy?” The answer is not immediately clear.  What about the friend of my frenemy, or the frenemy of my enemy?

It struck me that more work on this term was needed. And it struck me that the proper arena for rigorously standardizing such manipulations of interpersonal relationship terminology was mathematical. Specifically, the field of group theory suits such a task perfectly. 

In pursuance of this line of thought, I have written a research article on the subject, to which I link here:

My paper, "Frenemy Isomorphisms and Related Results"

My paper, "Frenemy Isomorphisms and Related Results"


In it you can find the details of this work, but I would like to summarize some of the main results here for the lay reader, and lay open the exciting conclusions my preliminary work suggests to a broader, non-technical audience.

Let us begin with an example we know and understand well; let us restrict ourselves to the world of friends and enemies.  Consider a map, or assignment, that associates to ‘friend’ the number 1 and to ‘enemy’ the number -1.  This assignment relates the friend-enemy dynamic to the algebraic structure of multiplication.  For example, who is the friend of my enemy?  ‘Friend’ is assigned to 1 and ‘enemy’ to -1, 1 x -1=-1, hence the friend of my enemy is my enemy.  This reflects our previous understanding of friend-enemy dynamics.  The equation -1 x -1 = 1 reflects the famous maxim “the enemy of my enemy is my friend.”  The algebraic structure perfectly describes the interpersonal dynamics.

In my paper I consider extensions of this framework to the friend-enemy-frenemy situation.  I conclude that there are two ‘frenemy structures’ that could work.  One (the Z4-structure) corresponds to assigning ‘friend’ to 1, ‘enemy’ to -1 and ‘frenemy’ to the imaginary number i.  In this case the frenemy of my frenemy can be calculated: i x i = -1.  The frenemy of my frenemy is my enemy.

In Section 4 of my paper the two possible frenemy structures are described via the following tables.  These tables are read as follows: to find what the frenemy of my friend is, find the entry in the row beginning with ‘frenemy’ and the collumn beginning with ‘friend’.  For either structure we see that the frenemy of my friend is my frenemy.

Model 1, the Z4 model
  friend frenemy enemy ?
friend friend frenemy enemy ?
frenemy frenemy enemy ? friend
enemy enemy ? friend frenemy
? ? friend frenemy enemy

Model 2, the Z2xZ2 model

  friend frenemy enemy ?
friend friend frenemy enemy ?
frenemy frenemy friend ? enemy
enemy enemy ? friend frenemy
? ? enemy frenemy friend

There are several striking implications of this work, and it indicates some problems that should be addressed.  First, the tables show that the frenemy of my frenemy is never my frenemy (Theorem 4.1 in my paper).  He or she is either my friend (in Model 1) or my enemy (in Model 2).  Thus, to determine which model actually reflects the friend-enemy-frenemy relationships, we need to determine who the frenemy of my frenemy is.  Model 2 seems more likely to me, but such relational analysis is a bit beyond my expertise as a mathematician, so I solicit your help.  Is the frenemy of my frenemy my friend, or my enemy?

Second, my work shows that there is no algebraic structure for the friend-enemy-frenemy dynamics that has only three objects (Lemma 3.2).  In either structure there is a fourth element, suggesting that the friend-enemy-frenemy categorization is incomplete!  There must be fourth, as yet undiscovered, relationship category.  I have denoted this category by ? in the tables above.  (Lest you be incredulous of this claim, I note that my procedure here is analogous to that which allowed Murray Gell-Mann, after noting the algebraic structure underlying known elementary particles, to predict the existence of new types of quarks.)  The behavior of this new category will be described by the proper table above.   For example, if the first table represents the correct model for the friend-enemy-frenemy relationships, we must look for a relationship category so that the enemy of my ? is my frenemy and so that the ? of my frenemy is my friend.  Again, I solicit your help. Perhaps this relationship is the enemend?  How can we understand this new category?


2 Responses to “Frenemy isomorphisms”

  1. 1 Adam November 23, 2008 at 2:01 pm

    I think the problem lies in the ambiguity of the term “frenemy”. According to the Wikipedia article, a frenemy can either be a friend that appears to be an enemy or an enemy that appears to be a friend. Maybe the relationships would be clear if you broke those out into two separate categories.

  2. 2 Mystic Rhythm February 17, 2010 at 7:23 pm

    I woke up this morning wondering the same thing, and came across your article. Now this begs the question, is the missing relationship a “friend that acts like an enemy”? Let’s call it an “enefriend” for the purposes of argument. That would seem to make sense within the Z4 and Z2xZ2 structures. The physical existence and practicality of an “enefriend” are a whole different discussion, but still, an interesting thought exercise.

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