I found an author who likes to use the phrase "$\underline{\textbf{onto over}}$" For example, the textbook author might write:
The set $X$ projected $\underline{\textbf{onto over}}$." the variable names $N$
It is common knowledge that the word "$\underline{\textbf{onto}}$" is often used to mean that a relation is surjective. For example, we might have:
$f$ is a mapping from set $A$ $\underline{\textbf{onto}}$ set $B$.
However, the set $X$ in my earlier example is not a relation. As such, $X$ cannot be surjective. Perhaps the author meant that the $\underline{\textit{projection}}$ relation is surjective?
We define an "$\underline{\textbf{evaluation}}$" to be a mapping $\mathrm{EVAL}$ such that there exists a set of finite-length character strings such that $\mathrm{EVAL}$ maps the strings to $\mathbb{R}$.
For example, the following is an evaluation:
EVAL1 = {{"$x$", 1}, {"$y$", 2}}
The particular text that I am reading has given the word "$\underline{\textbf{over}}$" a special meaning. We define "$\underline{\textbf{over}}$" such a set of evaluations $\mathrm{EVALS}$ is "$\underline{\textbf{over}}$" a set $\mathrm{VAR\_NAMES}$ if for all $\mathrm{EVAL}$in $\mathrm{EVALS}$, $\mathrm{EVAL}$ is a mapping from $\mathrm{VAR\_NAMES}$ to $\mathbb{R}$.
Consider the following example:
We have variables named "$x$" and "$y$"
EVAL1 = {{"$x$", 1}, {"$y$", 2}}
EVAL2 = {{"$x$", 111}, {"$y$", 222}}
EVALS = {EVAL1, EVAL2}
For our example, the set $\mathrm{EVALS}$ is $\underline{\textbf{over}}$ the set of variables names $\{$ "$x$", "$y$" $\}$
The above concludes the example, so the follows $\mathrm{EVALS}$ is not necessarily the same as that used in the example. Suppose that $\mathrm{EVALS}$ is a set of evaluations over a set of variable names $\mathrm{ALL\_VARS}$. Suppose that $\mathrm{SOME\_VARS}$ is a subset of $\mathrm{ALL\_VARS}$. What might the following mean?
$\mathrm{PROJ\_EVALS}$ is $\mathrm{EVALS}$ projected "$\underline{\textbf{onto over}}$" $\mathrm{SOME\_VARS}$
The author does not define "projection;" the author uses the term "projection" as if we are already supposed to know what it means. I am familiar with projection from linear algebra and elsewhere, but I am not sure what it means here.