I have this very simple equation for arbitrary $a,b,c,d,e,f\in\mathbb{R}$ with $\mathbb{R}$ endowed with the usual order :
$$a+b+c=d+e+f$$
So, if this equality is verified, I know that
$$\left\lbrace a-d\right\rbrace =\left\lbrace e-b\right\rbrace +\left\lbrace f-c \right\rbrace\quad (*)$$
And consquently that
$$\text{if }e\geq b\text{ and }f\geq c\text{ then }a\geq d\quad (**)$$.
But all rearrangements of $(*)$ give a statement in the design of $(**)$ :
We can exchange $a$ in the left-hand side with $b$ or $c$ in the right-hand side, we get $3$ rearrangements. The others follow by exploiting the symmetries in $(*)$. Firstly, the braces can be switched rearranging the signs, multiplying by $3$ the number of rearrangements. Secondly, terms in the right-hand side can be permuted, multiplying by $2$ the number of rearrangements. Thirdly, sign can be reversed on both sides multiplying by $2$ the number of rearrangements. After all, there are $36$ rearrangements that generate $36$ statements in the design of $(**)$.
I denote by $\pi,\sigma,\ldots$ permutations in the set of permutations of $3$ distinct objects.
I can deduce that $(*)$ implies that for all $\pi,\pi^{\prime},\sigma,\sigma^{\prime}$ permutations of $3$ distinct objects
$$\text{if } \pi(e)\geq\sigma(b)\text{ and } \pi(f)\geq\sigma(c)\text{ then }\pi(a)\geq\sigma(d) \quad (\bullet)$$
and
$$\text{if } \pi^{\prime}(b)\geq\sigma^{\prime}(e)\text{ and } \pi^{\prime}(c)\geq\sigma^{\prime}(f)\text{ then }\pi^{\prime}(d)\geq\sigma^{\prime}(a) \quad (\bullet \bullet)$$
So my question is very simple in fact but difficult for me, if I suppose that $(\bullet)$ and $(\bullet\bullet)$ is true it is possible to show that
$$a+b+c=d+e+f$$
is true ?
if anyone knows the algebraic books that I should see ?
Consider $(a,b,c,d,e,f)=(0,0,2,1,1,1)$. These numbers satisfy both sets of your order-based conditions (in most cases trivially, due to the premise of the implication being false), but $a+b+c\not= d+e+f$.
Very informally speaking, conditions based purely on comparison between some quantities are unlikely to be strong enough to say anything about their arithmetical properties (they might imply equality of the quantities involved though).