Suppose I have an ordered set $X$ with a finite number of elements. Say, for instance, $S = \{0.16, 0.34, 1.15, 3.16, 5.7\}$. Note that for "ordered set" I mean that the set elements are ordered starting from the smallest to the biggest. First question: is it correct to call such a set an "ordered set"?
Also, suppose that I partition set $S$ in such a way that I do not alter the order of its elements. Some admissible partitions for example could be $\{\{0.16, 0.34\}, \{1.15, 3.16\}, \{5.7\}\}$ or $\{\{0.16, 0.34, 1.15\}, \{3.16, 5.7\}\}$. Second question: how do I call such partition of an "ordered set"?
Concerning the second question, I found the term "composition" or "ordered partition", but I am not sure it is correct.
A set is a simple collection of unique elements. Therefore when you want to define some kind of relationship between elements in that set, you need to define what's called a relation. What you commonly refer to as "order from smallest to biggest" is in fact the most naïve order binary relation that you can think of. So what is an ordered set ?
An ordered set is a set that has at least a partial order. Which basically means that some of the elements are comparable. A totally ordered set is a set in which all elements are comparable.
I think what you mean by "ordered set" is "sorted set" but it makes little sense because elements in a set aren't meant to be arranged in a particular order. If you want the elements in your set to have a certain "sequence" it will no longer be a set but rather an indexed family which the order in the family is defined by a function. This is refered to in computer science as an array. I think this is what you are looking for.
As for the second question I'm not sure what you are refering to but you could always define a relation for a set S of set of real numbers that looks like this :
$U \leq_{S} V : \max(U) \leq_{\mathbb{R}} \max(V)$
With $\leq_{\mathbb{R}}$ being the usual order relation on the real numbers.