Double Ducky (or Fu-two-shiki)

Question

There are two* logical, discrete ways to solve this Futoshiki puzzle, each with its own unique solution. You can solve it using the letters given, or use the numbers 1-5, but if you map the letters to numbers, there must be a justification for how you map them (ie. you can't arbitrarily assign a letter to a number). The rules of Futoshiki are:

1. Each row and column must contain one of each digit from 1-5 (or letter in "DUCKY").
2. All inequality constraints between boxes must be satisfied (ie. 1 < 5; you'll have to figure out what the comparison between letters is depending on which route you take).

Here is a penpa-edit (without the letters entered in case you want to use numbers instead), and here is a transcription:

[ ] [ ] [D] [ ] [ ]  
 v   v           ^   
[ ] [ ] [U]<[ ] [ ]  
                     
[D] [U] [C] [K] [Y]
 v                   
[ ] [ ] [K] [ ]<[ ]
                 v   
[ ]>[ ] [Y] [ ]<[ ]

A full answer of both solutions is preferable, but if you want to submit a partial answer of only one solution, I will accept the other solution (since it was left to last, it seems that it was harder), if that makes sense.

*Technically, there are three, but two of these methods are simply "inverses" of each other, so only one of them will count.

Answer 1

First Solution

We can try the following

Assign the letters to numbers depending on the order in which they appear in the alphabet (C=1, D=2, K=3, U=4, Y=5)

Step 1

First we can fill in the Cs in a unique way.

Step 2

The inequalities in the bottom right force some letters which gives us some more of the rest of the grid.

Step 3

Noticing that the top of the fourth column must be a U we find that the rest of the grid solves quite quickly.

Second Solution

I have also noticed that

If we look at the Scrabble scores of D,U,C,K,Y we get D=2, U=1, C=3, K=5, Y=4, so maybe this will lead to a solution

Step 1

We can fill in all of the Us quickly

Step 2

The bottomright corner then resolves pretty quickly (the last two squares in the right column cannot contain K or Y so must contain C and D) and we can fill in a bit more of the grid using that.

Step 3

Now the bottom left is filled in in just one way and we can easily finish from here.

Hence, we have two valid and different solutions motivated by different aspects of the letters.