Sexistential

I'm looking for a word that describes a property P that possesses this attribute:

If object X is a set is composed of other objects, then:

1) P(X) => [P(y) for all y belonging to X]

or viceversa:

2) [P(y) for all y belonging to X] => P(X)

This attribute could also be:

3) P(X) => [P(y) for some y belonging to X]

or viceversa:

4) [P(y) for some y belonging to X] => P(X)

I'm sure this has been treated extensively in Formal Languages, epistemological treatises, philosophical treatises, or such. But in my ignorance I allow myself to define them. Let's see. Let's call them....

1) Integral
2) Essential
3) Existential
4) Diffusionary

Of course, these words already exist (except for, questionably, "Diffusionary"), and they would be easily confused if mixed in with some other kind of text, even if the reader knew these newly-born definitions. To attempt a unique linguistic distinction, I'll rename them to:

1) Xintegral
2) Xessential
3) Xexistential
4) Diffusionary

because Google seems to think that "diffusionary" is inexistent, or at least impopular. The pronunciation of these, I'll leave to the reader (since English is so phonetically ambiguous anyway), but I suggest a slightly accentuated "S" for the initial X's, maybe like the "X" in "Xerces" as is pronounced in the acclaimed "300" movie. Note also the meaningless innuendo created in the 3rd word.

So I postulate that, in the realm of the physical (in a simplified fashion):

1. "Small" is xintegral.
2. "Big" is xessential.
3. "Hot" is xexistential.
4. "Impure" is diffusionary.

The objective of these definitions comes from a want to ponder about questions such as:

Is Beauty xintegral/xessential/xexistential/diffusionary? Evil? True? Good? Blue? Sharp? Significant?

And since I'm on the topic, I might as well define "sexistential".

Sexistential (adj): Property of something that relates to sexual activities, or to the lack thereof, to a degree such that it motivates existentialist thoughts and feelings.