Consider a finite set $X = \{x_1,\ldots,x_n\}$ whose elements are totally ordered. In fact, for concreteness, assume that $X$ is a set of reals, and without loss of generality assume that $x_i$ is the $i$th element in the sorted list of elements, i.e. $x_1 \leq x_2 \leq \ldots \leq x_n$. Let $
My question is: what is the lattice generated by this partial order, and how can we characterize it? Is it well known, or isomorphic to something more familiar? Please forgive the vagueness of the question: I am primarily interested in learning what (if anything) this thing is called in combinatorics, and where I could learn more about it.
Thanks in advance!
PS: I am ultimately even more interested in another lattice defined by $I \subseteq J$ iff $\displaystyle\sum_{i\in I}x_i \leq \displaystyle\sum_{j\in J}x_j$, which clearly contains the first as a sub-lattice. I asked about the former since it seems simpler in some sense, but any remarks about either are quite welcome.