Converses of transitivity

Let $\prec$ be a binary relation. Two logically-equivalent ways to express transitivity are \begin{align} & a \prec b \to \forall x (x \prec a \to x \prec b) \\ & a \prec b \to \forall x (b \prec x \to a \prec x) \end{align} They have two distinct converses: [1] \begin{align} & \forall x (x \prec a \to x \prec b) \to a \prec b & \text{A} \\ & \forall x (b \prec x \to a \prec x) \to a \prec b & \text{B} \end{align}

The only reference I've found to them is [2], which calls them "converse transitivity from below" and "converse transitivity from above", adding:

Converse transitivity from below and converse transitivity from above are not necessarily the same, as the spouse/sibling example shows, and it is possible to construct other examples in which one or the other or both conditions do not hold.

My questions are the following:

1. Do properties A and B have other names in the literature?
2. Are they discussed in greater detail elsewhere in the literature?
3. Is there a name for a binary relation that satisfies transitivity and both of its converses?
4. What are some common, general mathematical structures that satisfy the above?

1. Note that both imply reflexivity.

2. John Randolph Lucas. §9.9 Converse Transitivity. Conceptual Roots of Mathematics. Routledge, 2002-09-11. Pages 253-254.

Solution 1:

Both of these converses are in fact just equivalent to reflexivity. You have already observed that they imply reflexivity, since the hypothesis of either converse is always true if $a=b$. Conversely, suppose $\prec$ is reflexive and for all $x$, $x\prec a$ implies $x\prec b$. In particular, we know that $a\prec a$ by reflexivity, and therefore $a\prec b$. The other converse works similarly.