All of these are shortened versions of longer phrases, which we no longer use (and which only a few actually *know). For example, the ubiquitous "goodbye" is actually a greatly shortened version of "God be with ye (you)".

So something like "Regards" might be short for

Sending my best regards to you and your family.

or

This letter is sent with my regards for you and your family.

As with "goodbye" these expression grow shorter and shorter over time until today we sign off with barely two letters: "C U"

The grammar doesn't really matter, only what the expressions represent to the reader. "Your truly", for example, hardly means that I consider myself a devoted servant of the reader. It's just a polite expression to indicate I don't actually hate them.

All of these are shortened versions of longer phrases, which we no longer use (and which only a few people might actually know). For example, the ubiquitous "goodbye" is actually a greatly shortened version of "God be with ye (you)".

So something like "Regards" might be short for

Sending my best regards to you and your family.

or

This letter is sent with kind regards toward you and your family.

As with "goodbye" these expression grow shorter and shorter over time until today we sign off with the briefest possible: "C U"

The grammar doesn't really matter, only what the expressions represent to the reader. "Yours truly", for example, hardly means that I consider myself a devoted servant of the reader. It's just a polite expression to indicate I don't actually loathe them.