  In this paper, we define partial orders (POs) of polar codes based on the
Bhattacharyya parameter and the bit-error probability, respectively. These POs
are applicable to arbitrary binary memoryless symmetric channel (BMSC).
Leveraging the extremal inequalities of polarization transformation, we derive
new POs for BMSC based on the corresponding POs observed in the Binary Erasure
Channel (BEC). %Additionally, we discover more special POs in the Binary
Symmetric Channel (BSC). We provide examples that demonstrate the inability of
existing POs to deduce these novel POs. Furthermore, we establish upper bounds
for the expansion parameter $\beta$ if the polar codes constructed by
$\beta$-expansion method obey these POs.
