There does exist what we call "rootless" voicings in harmony. These are chords in which the root is implied by the upper harmonies. Typically, the 3rd and the 7th are the primary indicators of chord quality, and the 5th is secondary. Rootless voicings are most commonly used in settings where an instrument such as piano or guitar is providing harmonic support in an ensemble/group setting, and an instrument (usually bass) has already stated the root of the chord. In composition, rootless voicings are utilized to keep the tonal center ambiguous and allow for the possibility of shifting the root of the chord underneath the harmony to create interest. Here are some examples of rootless voicings:
If you were to play the rootless voicings in the above picture without their root, the placement and use of the chord tones would imply the root. Why does this work?
As I said earlier, the 3rd and 7th give the strongest indicator of the chord quality, and the 5th & 9th are the usual secondary indicators. With that in mind, chord quality can be determined without a root when the following tones exist within the chord:
Major => Maj. 3rd & Maj. 7th
Minor => Min. 3rd & Min. 7th
Minor Major => Min. 3rd & Maj. 7th
Dominant => Maj. 3rd & Min. 7th
Diminished 7 => Min. 3rd, Min. 5th, Dim. (bb) 7
Half Diminished => Min. 3rd, Min. 5th, Min. 7th
Augmented Major => Maj. 3rd, Sharp 5th, Maj. 7th
Augmented Dominant => Maj. 3rd, Sharp 5th, Min. 7th
Experiment with playing these chords & intervals without the root first, and then with the root -- you will become more familiar with the idea of not relying upon the root itself to express the tonal center of the chord. Good luck!