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 welldefined, 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"
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, nontechnical 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 friendenemy 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 friendenemy 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 friendenemyfrenemy situation. I conclude that there are two ‘frenemy structures’ that could work. One (the Z_{4}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 Z_{4} model

friend 
frenemy 
enemy 
? 
friend 
friend 
frenemy 
enemy 
? 
frenemy 
frenemy 
enemy 
? 
friend 
enemy 
enemy 
? 
friend 
frenemy 
? 
? 
friend 
frenemy 
enemy 
Model 2, the Z_{2}xZ_{2} 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 friendenemyfrenemy 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 friendenemyfrenemy dynamics that has only three objects (Lemma 3.2). In either structure there is a fourth element, suggesting that the friendenemyfrenemy 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 GellMann, 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 friendenemyfrenemy 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?
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