Sentence ending with こと or もの in mathematics definitions

I'm reading a mathematics textbook, and there are a number of sentences which end with もの or こと.

I can work out the intended meaning no problem, so what I would like explained to me, are the rules for when I can use this grammatical construction.

This seems to happen in definitions, especially if mathematically written conditions are involved.

For example:

すなわち

1. m1, m2 ∈ N ⇒ m1 + m2 ∈ N,　さらに 0 ∈ N

2. r ∈ R, n ∈ N ⇒ r•n ∈ N

となるもの。

or

f: S → T 全射 (surjection, epimorphism) であるとは、f (S) = T が成立すること。

I asked a (non-Japanese) mathematician, who told me that this can be used to give commands, e.g., 勉強すること。However, I don't see why a command would appear in definitions in this way...

The sentence-final copula である ("be") is almost always omitted because it's obvious in definitions, leaving the sentences looking like ending with nouns. Both もの and こと are frequently used nominalizers translating "what do ~" and "doing ~" respectively.

すなわち、1. (...) 2. (...) となるもの。
i.e. what satisfies 1. (...) and 2. (...).

f:S→T が全射であるとは、f(S)=T が成立すること。
f:S→T is a surjection means that f(S)=T holds true.
less literally, f:S→T is a surjection when f(S)=T holds true.

I looked up "成立" and two of the meanings are "coming into existence; conclusion" while having the meaning "to hold true" as a suru verb. I think "成立すること" is simply to address the multiple meanings of the word, and it also refers to the noun (f(S)=T) possessing the quality of the verb (成立する) by nominalizing the verb (こと).

I haven't read a math book in a while, but if I can still remember mathematicians parlance (haha) I would translate it as: "When f:S→T is a surjection, f(S)=T still holds true."

• I should have maybe given more context. The examples come from definitions. I would translate the second one as: "A morphism f:S→T is an epimorphism if f(S)=T holds true." Its from the definition of an epismorphism. – James May 30 '15 at 14:14