The word ligature can mean more than one thing. Its primary sense is of something that ties, binds or constricts. The word can be used in this generic sense or more specifically in certain contexts.
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Ligaturing is a behavior encoded in fonts: if a modern font is asked to display "h" followed by "r", and the font has an "hr" ligature in it, it can display the ligature.
No, the "ct" ligature is another example of a ligature of Latin letters commonly seen in older type styles.
As for the case of the "hr" ligature, display of a ligature is a matter for font design, and does not require separate encoding of a character for the ligature.
Ligatures (special combined characters, "fi" "fl" "ff" and "ffi" being the most common) were originally created for two reasons.
However--the ligature is really not necessary or important because the "f" and the "i" don't collide in either the Roman or the Italic.
The f and i are in no danger of collision, and the "fi" ligature not only merges the two for no good reason, it pulls the top of the "f" over the right (where it wasn't before) and keeps the dot on the "i" which just ends up making the thing look crowded.