This topic is about the way that you relate to dream characters, and perhaps scenes and objects as well. Sorry, it's not about romance (but you don't want my advice on that anyway ;). I place it in Dream Control because it mainly concerns archetypes. It can be a lesson on creating characters, or interpreting them, or introspecting into yourself.

Algebra

In mathematics, algebra is about describing relationships. There are two essential elements: symbols and balance. Typically, you have an equation with some variables. For example: a + b = c. The variables are unknowns. The equality symbol declares that the two sides must be balanced. Algebra is a tool with many uses. If there is only one unknown, then you can "solve for the unknown" (e.g. x + 2 = 3). If there are many unknowns, then they describe a fixed relationship (e.g. y = 2x + 1). In that latter example, you can't solve for x or y separately, but you can plot the relationship as a line. Things get even more interesting when you have multiple equations or balances besides equality, but let's focus on the fundamentals.

If you take away nothing else, let it be this: Algebra is a tool and tools do not exist for their own sake but as a component of the process of understanding. Understanding is propelled by asking questions.

Translation

For dream characters, the symbols are archetypes and the balance is relationships. This isn't about math anymore, but there will still be some numbers. A relationship has two parties: you and them. That relationship can be either positive or negative. It can be symmetrical or asymmetrical. If there is asymmetry, then we can identify the two halves separately. And finally, we must remember that this is not math or logic. So we must consider that our seemingly strict relationships are actually ambiguous or non-existant. To borrow from logic, this system is "fuzzy."

Classification

I identify 8 classes. Because there is cosmic irony and because of my last point about always doubting that your system is perfect, there are probably more or less than 8. Nonetheless, I enumerate them:

Symmetrical, positive
Archetype: Friend
Alternates: Partner, Comrade, Mate

Symmetrical, negative
Archetype: Rival
Alternates: Foil, Nemesis, Doppelgänger

Asymmetrical, positive
Archetype A: Teacher
Archetype B: Student
Alternates: Guardian-Ward, Mentor-Protégé

Asymmetrical, negative
Archetype A: Master
Archetype B: Slave
Alternates: Manipulator-Puppet, Elite-Outcast

Combination/Ambiguous
Archetype: "It's Complicated"
Comments: A relationship may be complex. It may be a combination of others, possibly in unequal proportion. It may also vary through time or context.

Neither/Nor
Archetype: Stranger
Comments: A relationship is not fixed. It is discovered as details are revealed. Perhaps previously unknown elements are discovered even later.

Application

So that's kinda neat. It's a system for classifying characters. But how to use it? The first thing to note is that reality (even dream reality) is not so neat. Consider, for example: how would you classify a brother? Is your brother a friend or a rival? Maybe he's much younger or older than you, so you are on either side of a guardian-ward relationship. Consider a mother-daughter relationship. You might assume it is asymmetrical, either positive or negative. But over time, maybe it evolves into a symmetrical friendship or something else.

Thus, I suggest this system is not so much about classification but experimentation. Look at your actual relationships with dream characters and see if you have mis-classified them because you rely too much on past beliefs and not enough on current context. As I hinted above, this is a process and it starts with asking questions.

Questions to Explore

Do you have recurring characters inspired by real people? Have you perhaps mis-classified your mother, father, sister, brother because you put them in the wrong bucket? Do you see your real-life teacher as an archetypal teacher as as slavedriver?

Does a rival need to be entirely negative? If they are a foil, maybe you have more in common.

Does a relationship depend on context? Is your boss at work different from your boss at a cocktail bar after work?

Does a dream guide have to be the teacher in a teacher-student pair? Could it be a friend? Or always a stranger?

When you meet a stranger, do you immediately classify them? Do you see every stranger as a potential rival? What does that reveal about you?

Take the point of view of the character. Would they classify your relationship the same way? You might seem them as teacher-student, but their attitude is master-slave. Remember that dream characters are you. So what does that incongruity reveal about you?

What is your relationship with your dreamlife? Are you the master and dreams your slave? Or is it your teacher and you the student? Could you be friends?

If you reframe your understanding of an existing relationship, what might change? Might it "solve for the unknown?" Might you grow or learn from it?