Heterosexuality and homosexuality have commonly understood definitions in western society. For reasons that will become clear, let’s refer to this as Strong Heterosexuality and Strong Homosexuality.

For historical reasons, Strong Heterosexuality refers to a relationship consisting of exactly one man and one woman. Some proponents of Strong Heterosexuality include other descriptors like requiring it to be “for the purposes of procreation”, but this is out of scope as this article only considers the classification of relationships. Similarly proponents of Strong Heterosexuality may require the participants to be cisgender but we will not be considering that aspect of gender in this article.

Strong Heterosexuality’s counterpoint, Strong Homosexuality, covers every other relationship. For example, a relationship consisting of two men or two women would be Strongly Homosexual, but relationships between a man and a nonbinary person, or 2 non-binary people, would be considered Strong Homosexual.

But this reveals a weakness in the naming of Strong Heterosexuality and Strong Homosexuality. The hetero- prefix indicates that the participants are different where the homo- prefix indicates that participants are the same (in terms of gender). But some relationships, such as the previously mentioned relationship between a man and a non-binary person are considered Strongly Homosexual despite consisting of people of two different genders.

Weak Heterosexuality

Suppose we define Weak Heterosexuality as relationships consisting of no identical genders. Suppose $ G $ is the set of all genders. A Weakly Heterosexual relationship a set of genders $ H $ such that $ H \subset G $ and $ |H| = 2 $ and $ \forall g_1, g_2 \in H, g_1 \neq g_2 $. Note $ | G | \geq 3 $¹ so $ H $ is always a proper subset and there is always an $ n \in G $ such that $ n \ni H$.

Any relationship that is Strongly Heterosexual is also Weakly Heterosexual by definition.² However a relationship between a man and a non-binary person would be Weakly Heterosexual because $ man \neq nonbinary$.

But non-binary is not really a gender but a classification of genders: $ NB \subset G$. Unfortunately the definition of $ NB $ is multifaceted and hotly debated so we cannot give a full specification. However we can at least assert that $ man, woman \ni NB $

Weak Homosexuality

It is easy to derive Weak Homosexuality from Weak Heterosexuality: A Weakly Homosexual relationship is a set of genders $ Q $ such that $ Q \subset G$ and $ |Q| = 2$ and $ \forall g_1, g_2 \in Q, g_1 = g_2$.

So any relationship consisting of two people with exactly the same gender is Weakly Homosexual. Otherwise it is Weakly Heterosexual.

It is worth noting that this is not necessarily a common view: this author has heard the claim that (paraphrasing) “I’m non-binary so all my relationships are homosexual,” but in fact they would only be Strongly Homosexual and would only be Weakly Homosexual if they only dated people with the same non-binary gender.

Annex: Polyamory

The above definitions of Weak Heterosexuality and Homosexuality include the requirement that the set of genders in the relationship have only two genders. But this requirement is unnecessary. Polyamorous relationships consisting of more than two people can still be examined under the framework of Weak Heterosexuality and Homosexuality.

For example a lesbian polycule consisting only of women would be Weakly Homosexual as all members have identical genders.

However if a non-binary person joined the same polycule, a different gender would have been introduced and the polycule would be Weakly Heterosexual.

Polyamorous relationships cannot be Strongly Heterosexual and are always Strongly Homosexual.

Annex: Genderfluid

Some people identify as genderfluid and have dynamic genders. Suppose $ f(t) \in G $ represents a person’s gender for some time $ t $³. If $ f(t) $ were in a relationship with $ g \in G $ their relationship would be Weakly Homosexual $ \forall t, f(t) = g $ and Weakly Heterosexual for all other $ t $.

If $ f(t) $ is not defined for all $ t $ see Annex: Agender.

Annex: Agender

Some people identify as “agender” and do not have a gender. This could be defined such that, if $ g $ is an agender person’s gender, then $ g \ni G$. But agender people do not have gender so they cannot have $ g $ such that $ g $ is their gender.

Let us examine the definition of Weakly Heterosexuality more carefully. Weak Heterosexuality implicitly assumes that a relationship is a set $ R \subseteq P $ where $ P $ is the set of all people. Thus $ H = \lbrace \forall p \in P : gender(p) \rbrace $. However $ gender(p) $ is undefined for agender people and if a relationship involves an agender person then $ |H| < 2 $. The definition of Weak Heterosexuality is undefined when $ |H| < 2$ so relationships consisting of one person with a gender and one agender person are neither Weakly Heterosexual nor Weakly Homosexual.

Polyamorous relationships could involve three people with gender and any number of agender people and the orientation of the relationship would be defined by the genders of the three people with gender and the agender people would not contribute to the definition of sexual orientation of the relationship.