  Despite the insights gained in the last few years, our knowledge about the
formation and evolution scenario for the spheroid-dominated galaxies is still
incomplete. New and more powerful cosmological simulations have been developed
that together with more precise observations open the possibility of more
detailed study of the formation of early-type galaxies (ETGs). The aim of this
work is to analyse the assembly histories of ETGs in a $\Lambda$-CDM cosmology,
focussing on the archeological approach given by the mass-growth histories.We
inspected a sample of dispersion-dominated galaxies selected from the largest
volume simulation of the EAGLE project. This simulation includes a variety of
physical processes such as radiative cooling, star formation (SF), metal
enrichment, and stellar and active galactic nucleus (AGN) feedback. The
selected sample comprised 508 spheroid-dominated galaxies classified according
to their dynamical properties. Their surface brightness profile, the
fundamental relations, kinematic properties, and stellar-mass growth histories
are estimated and analysed. The findings are confronted with recent
observations.The simulated ETGs are found to globally reproduce the fundamental
relations of ellipticals. All of them have an inner disc component where
residual younger stellar populations (SPs) are detected. A fraction of this
inner-disc correlates with bulge-to-total ratio. We find a relation between
kinematics and shape that implies that dispersion-dominated galaxies with low
$V/\sigma_L$ (where $V$ is the average rotational velocity and $\sigma_L$ the
one dimensional velocity dispersion) tend to have ellipticity smaller than
$\sim 0.5$ and are dominated by old stars. Abridged

Astrophysics of Galaxies;Cosmology and Nongalactic Astrophysics