Does it sometimes denote, in measurement theory; often the qualitative counterpart to $\geq $ in the numerical representation; when one wants to numerically represent a totally ordered qualitative probability representation:

$$ A ≽ B \leftrightarrow A > B \leftrightarrow F(A) \geq F(B)$$.

**On** the other hand. I have seen it used in multi-dimensional partial or even total, orderings under the "ordering of major-ization" for vector valued functions or for a system for two kinds of orderings.

One for (numerical) ordinal, $>$ comparisons and another, $≽$ for (numerical) differences, sums, or a some other kind of relation, to fine grain, the representation to ensure (or some kind of) unique-ness, rather than merely strong represent-ability.

See Marshall

*Marshall, Albert W.; Olkin, Ingram*, Inequalities: theory of majorization and its applications, Mathematics in Science and Engineering, Vol. 143. New York etc.: Academic Press. XX, 569 p. \$ 52.50 (1979). ZBL0437.26007.

Such functions or representations, may use both the curly greater $≽$ than and $\geq $ in the numerical or functional representation and are compatible with total orderings.

$$a,b\in \Omega^{n}\; a <b \,,\quad \text{or}, \quad, a= b, \quad \text{or}\quad a < b$$

where one cannot usually adding up distinct
$$a_1 \in \Omega_i\,; b_j\in \Omega_j; j \neq i$$

$$a_3 \in \Omega_c;\, ; \, P(a_i) + P(b_j) = P(a_3)$$.

Sometimes I "(conjecture)" that $≽$ denotes a weaker version of $(A)$ or $(A_1)$ (strong complementary add-itivity) which orders the differences and sums, is something over and above

$(B)$;strict monotone increasing/strong representability/order embedding- above reflecting and preserving .

$$(A)\,[f(x_1)+f(x_2)≽ f(y_1) + f(y_2)] \rightarrow [f(x) \geq f(y)] \text{where} \,; x=x_1 +x_2\,, y=y_1+y_2 $$
or
$$(A1)\,[x\, \geq y ]\,\rightarrow [f(x_1)+f(x_2)≽ f(y_1) + f(y_2)]\text{where} \,; x=x_1 +x_2\,, y=y_1+y_2 $$

Rather than:$$(B)$$

$$(B)\,x\, \geq y \leftrightarrow f(x) \geq f(y)\, \text{where} \,; x=x_1 +x_2\,, y=y_1+y_2 $$

In contrast to $(B)$ a standard order embedding (strict monotone increasing function)

Where in $(B)$ the numerical function, $F$ or representation is 'merely strong'. That is $F$ is a monotone strictly increasing function of some entity $x$ where $F(x)$ is the entity one wishes to order by $x$ .

Sometimes, even in a infinite and uniformly and non-atomic, continuous total order representation, nothing unique will come out. If its infinite /non-atomic in the wrong sense.

ie in a super-atomic or entangled lexicographic system. Where the system is dense/or continuum dense/non-atomic/bottom-less in the wrong sense. Such as a spin $1/2$ system in quantum mechanics, or a lexicographic entangled, system where its infinite in wrong sense, in the vertical, not in the horizontal, not within a basis of space/basis.

$(A)$ is arguably a lot stronger than $(B)$, or at least is, if relatively unrestricted.

Where $(A)$, may be used not to extend the ordering so much as much as make a total ordering 'numerically precise; to put some constraints on sums and differences, where the events are the not kinds of things that usually add up.

Say in an entangled multidimensional quantum spin 1/2 system or utility representation where the events are on complementary spaces and mixtures are not allowed, for example.

That is, to, put a metric on differences (say on a two outcome system). That is something stronger, than a, total or non atomic/dense order, where(merely) every event.

Even if the entire system is totally ordered within and betwixt the distinct $\Omega$. So that the system is solv-able, and uniquely so.