Characterization of Pomonoids by Properties of I-Regular S-Posets

In 2005, Shi defined I-regular S-posets and used this concept to characterize PP-pomonoids and po-cancellable pomonoids. In this paper, we continue to develop the homological classification of pomonoids by using the I-regularity of S-posets. First, we characterize pomonoids over which all I-regular S-posets have one of the properties around projectivity or injectivity, and many known results are generalized. Moreover, some possible conditions on pomonoids that describe when their diagonal posets are I-regular are found. Finally, some characterizations of pomonoids by the I-regularity of their Rees factor posets are given.