(Abridged abstract) We present an analysis of the X-ray emission of the rapidly rotating giant star YY Mensae observed by Chandra HETGS and XMM-Newton. Although no obvious flare was detected, the X-ray luminosity changed by a factor of two between the XMM-Newton and Chandra observations taken 4 months apart. The coronal abundances and the emission measure distribution have been derived from three different methods using optically thin collisional ionization equilibrium models. The abundances show an inverse first ionization potential (FIP) effect. We further find a high N abundance which we interpret as a signature of material processed in the CNO cycle. The corona is dominated by a very high temperature (20-40 MK) plasma, which places YY Men among the magnetically active stars with the hottest coronae. Lower temperature plasma also coexists, albeit with much lower emission measure. Line broadening is reported, which we interpret as Doppler thermal broadening, although rotational broadening due to X-ray emitting material high above the surface could be present as well. We use two different formalisms to discuss the shape of the emission measure distribution. The first one infers the properties of coronal loops, whereas the second formalism uses flares as a statistical ensemble. We find that most of the loops in the corona of YY Men have their maximum temperature equal to or slightly larger than about 30 MK. We also find that small flares could contribute significantly to the coronal heating in YY Men. Although there is no evidence of flare variability in the X-ray light curves, we argue that YY Mens distance and X-ray brightness does not allow us to detect flares with peak luminosities Lx <= 10^{31} erg/s with current detectors.
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