We performed a uniform and detailed analysis of 1112 high-resolution spectra of FGK dwarfs obtained with the HARPS spectrograph at the ESO 3.6 m telescope (La Silla, Chile). Most stars have effective temperatures 4700 K ≤ Teff ≤ 6300 K and lie in the metallicity range of -1.39 ≤ [Fe/H] ≤ 0.55. Our main goal is to investigate whether there are any differences between the elemental abundance trends (especially [α/Fe] ratio) for stars of different subpopulations. The equivalent widths of spectral lines are automatically measured from HARPS spectra with the ARES code. The abundances of three α elements are determined using a differential LTE analysis relative to the Sun, with the 2010 revised version of the spectral synthesis code MOOG and a grid of Kurucz ATLAS9 atmospheres. The stars of our sample fall into two populations, clearly separated in terms of [α/Fe] up to super-solar metallicities. In turn, high-α stars are also separated into two families with a gap in both [α/Fe] ([α/Fe] ≈ 0.17) and metallicity ([Fe/H] ≈ -0.2) distributions. The metal-poor high-α stars (thick disk) and metal-rich high-α stars are on average older than chemically defined thin disk stars (low-α stars). The two α-enhanced families have different kinematics and orbital parameters. The metal-rich α-enhanced stars, such as thin disk stars have nearly circular orbits, close to the Galactic plane. We put forward the idea that these stars may have been formed in the inner Galactic disk, but their exact nature still remains to be clarified. Tables 1 and 2 are available in electronic form at http://www.aanda.org