A binary relation $R$ over $D$ is semitransitive if the following condition holds for all $a, b, c, d\in D$:
(1) If $aRb$ and $bRc$, then $aRd$ or $dRc$.
A binary relation $R$ over $D$ is Ferrers if the following condition holds for all $a, b, c, d\in D$:
(2) If $aRb$, $cRd$, and not $cRb$, then $aRd$.
A semiorder $R$ is reflexive, semitransitive, and Ferrers. As usual, the strict part $P$ of $R$ is defined as $aPb$ iff. $aRb$ and not $bRa$, and the nonstrict part $I$ is defined as $aIb$ iff. $aRb$ and $bRa$. An interesting property of a semiorder is that while $P$ is transitive, $I$ is not transitive. Another definition for completenes: A binary relation $R$ is Euclidean iff. $aRb$ and $aRc$ implies $bRc$.
1. Is it true that $I$ is not Euclidean when it is derived from a total semiorder?
2. If I impose an additional constraint that $I$ must be Euclidean, does $R$ then become a preorder?
I'm asking because in preference theory and mathematical psychology, nice examples have been given for the non-transitivity of preferences (Luce's sugar in coffee example), and these lead to semiorder representations. But the case $aIb$, $aIc$ and $bPc$ is apparently never discussed in this literature and seems quite odd if $P$ is supposed to express preferences and $I$ is (non-transitive) indistinguishability.