The A-7 chord has the same notes as a C6 chord, the root of the A-7 is the 6th of the C6.
These chords are 'homonyms' of each other. In grammar, the words 'break' and 'brake' are homonyms -- they sound the same but mean something different. With these chords, the sound is the same, but the 'meaning' behind the construction and it's context in a song is different.
Sometimes a chord can be 'rootless' -- a fingering can imply a chord by virtue of an 'assumed root' -- that is, the root is not part of the chord's fingering, but we can assume the root based on the overall harmony the changes imply. These chords are great for comping, because the bassist is often filling in the root anyway.
Here is another 'chord homonym':
D-7b5 = Bb9 (no root) = F-6 = E+7b9
The same fingering for one can be used to imply another.
- If you use the b3 of a min7b5 as the root, you get a min6 chord.
- If you use the 13 of a min6 as the root, you get a min7b5 chord.
- If you use the b3 of a min7 as the root, you get a Maj6 chord.
- If you use the 13 of a Maj6 as the root, you get a min7 chord.
Plurality is different from inversions: Inverted chords are different fingerings of the same chord, with the same notes. Inverted chords just have a note other than the root as the bottom note. Plurality is where one fingering can be heard as multiple chords.