Regarding the "mathematical object $(X, s)$" described below (in general, or under some more specific conditions) I'd like to know whether it is called a particular name, for reference in the literature:
How do you call a set $X$ together with a map $s\!\!: X^2 \rightarrow R$
(0) in general, or
(1) under the condition that $\forall \eta \in X: s[ \, \eta, \eta \, ] = 0$, or
permitting that there may $\exists \eta, \phi \in X: s[ \, \eta, \phi \, ] \lt 0$, under the additional condition that:
(2) for any three elements $\rho$, $\sigma$ and $\tau \in X$
$(0 \geq s[ \, \rho, \sigma \, ] \geq s[ \, \sigma, \tau \, ] \geq s[ \, \tau, \rho \, ]) \implies $
$(s[ \, \rho, \sigma \, ] + s[ \, \sigma, \tau \, ] \geq s[ \, \tau, \rho \, ])$
?
A remark on Ãnequalities involving values $s[ \, \alpha, \beta \, ] \gt 0$ for suitable elements $\alpha$, and $\beta \in X$:
Considering 7 suitable elements of $X$, with corresponding 21 values
$s[ \, \omega, \kappa \, ] = 0$,
$s[ \, \omega, \lambda \, ] = 0$,
$s[ \, \omega, \mu \, ] = 0$,
$s[ \, \omega, \nu \, ] = 0$,
$s[ \, \mu, \rho \, ] = 0$,
$s[ \, \nu, \sigma \, ] = 0$, and
$s[ \, \kappa, \lambda \, ] \lt 0$,
$s[ \, \mu, \nu \, ] \lt 0$,
$s[ \, \kappa, \nu \, ] \leq 0$,
$s[ \, \lambda, \mu \, ] \leq 0$,
$s[ \, \kappa, \mu \, ] \gt 0$,
$s[ \, \lambda, \nu \, ] \gt 0$; and also
$s[ \, \rho, \sigma \, ] \lt 0$,
$s[ \, \omega, \rho \, ] \leq 0$,
$s[ \, \omega, \sigma \, ] \leq 0$, and moreover
$0 \geq s[ \, \kappa, \rho \, ] \geq s[ \, \omega, \rho \, ]$,
$0 \geq s[ \, \lambda, \sigma \, ] \geq s[ \, \omega, \sigma \, ]$, and finally
$s[ \, \kappa, \sigma \, ] \lt 0$,
$s[ \, \lambda, \rho \, ] \lt 0$,
$s[ \, \mu, \sigma \, ] \lt s[ \, \mu, \nu \, ] \lt 0$, and
$s[ \, \nu, \rho \, ] \lt s[ \, \mu, \nu \, ] \lt 0$,
then ("by construction") holds:
$s[ \, \rho, \sigma \, ] \leq s[ \, \kappa, \lambda \, ] \lt 0$.
Consequently, in case there exist suitable elements $\alpha$, and $\beta \in X$ for which
$s[ \, \kappa, \alpha \, ] \lt 0$,
$s[ \, \alpha, \lambda \, ] \lt 0$,
$s[ \, \mu, \beta \, ] \lt 0$,
$s[ \, \beta, \nu \, ] \lt 0$, and
$s[ \, \alpha, \beta \, ] \gt 0$,
$s[ \, \alpha, \omega \, ] \gt 0$,
$s[ \, \beta, \omega \, ] \gt 0$ such that
$\frac{s[ \, \alpha, \omega \, ]}{s[ \, \kappa, \lambda \, ]} = \frac{s[ \, \beta, \omega \, ]}{s[ \, \mu, \nu \, ]} = \frac{s[ \, \alpha, \beta \, ] + s[ \, \beta, \omega \, ]}{s[ \, \rho, \sigma \, ]}$,
then the following triangle inequality holds:
$s[ \, \alpha, \omega \, ] \leq s[ \, \alpha, \beta \, ] + s[ \, \beta, \omega \, ]$.
An example for a "space" (in the sense of a mathematical object) satisfying all these conditions may be any set of "events" which are pairwise related being "spacelike", or "lightlike", or "timelike" to each other, such that
any pair of "spacelike" events are characrerized by positive quasi-distance values among each other,
$(s[ \, \alpha, \beta \, ] > 0$, and$s[ \, \beta, \alpha \, ] > 0$,any pair of "timelike" events are characrerized by a (proper) duration value among each other, corresponding to
$0 \geq s[ \, \rho, \sigma \, ] = s[ \, \sigma, \rho \, ]$,for any pair of "spacelike" events there are events "lightlike" wrt. both, and
for any pair of "timelike" events there are events "lightlike" wrt. both, where
any pair of "lightlight" events would be assigned the value
$s[ \, \tau, \alpha \, ] = 0 = s[ \, \alpha, \tau \, ])$.
(This is an edited, third version of my question. The comments below, of January 2012, were based on a less organized first version. The second version, which had been published meanwhile and on which no further comments were issued, should hereby be considered withdrawn, as flawed.)
