On Flatness Properties And I-regularity Of S-posets

Let S be a pomonoid. In [14] it is proved that all strongly flat left S-posets are I-regular if and only if S is a left PP and left semiperfect pomonoid. The first part of this paper proves that left semiperfect property can be replaced by a weaker property （FP₂） or the left PP pomonoid can be extended to left PSF pomonoid. Furthermore, the notion of P-regularity is introduced and give the e-quivalent characterizations of condition （E） （strongly flat, projective, free） S-posets are P-regular. Secondly, some equivalent characterizations between flatness proper-ties and I-regularity of S-posets are obtained. The notion of C-SF-poset is intro-duced, and give the properties of pomonoids over which condition （E） （strongly flat, projective, free） S-posets are CSF posets.