We analyse the properties of the early-type dwarf galaxy population in the Hydra I cluster. We investigate the galaxy luminosity function (LF), the colour-magnitude relation (CMR), and the magnitude-surface brightness relation down to M_V~-10 mag. Another goal of this study is to find candidates for ultra-compact dwarf galaxies (UCDs). Two spectroscopic surveys performed with Magellan I/LDSS2 and VLT/VIMOS, as well as deep VLT/FORS1 images in V and I bands were examined. We identify cluster members by radial velocity measurements and select other cluster galaxy candidates by their morphology. One possible UCD candidate with M_V=-13.26 mag is found. Our sample of 100 morphologically selected dwarf galaxies defines a CMR that extends the CMR of the giant cluster galaxies to the magnitude limit of our survey (M_V~-10 mag). It matches the relations found for the Local Group and the Fornax cluster dwarfs almost perfectly. The Hydra I dwarfs also follow a magnitude-surface brightness relation similar to that of the LG dwarfs. Moreover, we observe a continuous relation for dwarf galaxies and giant early-type galaxies when plotting the central surface brightness mu_0 of a Sersic model vs. the galaxy magnitude. The effective radius is found to be largely independent of the luminosity for M_V>-18 mag, being R_e~0.8 kpc. We derive a very flat faint-end slope of the LF (alpha = -1.13 pm 0.04) from fitting a Schechter function, whereas fitting a power law for M_V>-14 mag gives alpha = -1.40 pm 0.18. Our findings suggest that early-type dwarf and giant galaxies are the same class of objects. The similarity of the dwarf galaxy scaling relations to other environments implies that internal processes could be more important for their global photometric properties than external influences. (abridged)
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