On a generalization of I-regularity

Abstract Let S S be a pomonoid. The projectivity and strong flatness of right S S -posets have been central topics in the homological classification of pomonoids in recent decades. In 2005, Shi et al. introduced I I -regular S S -posets and proved that all its cyclic S S -subposets are projective [Indecomposable, projective, and flat S-posets, Comm. Algebra 33 (2005), no. 1, 235–251]. Feng subsequently introduced CSF S S -posets, where all its cyclic S S -subposets are strongly flatness [CSF S S -posets and C(P) S S -posets, Master’s thesis, Northwest Normal University, Gansu, 2020]. In this article, we define a new class of strongly (SF)-cyclic S S -posets, which lies strictly between the classes of I I -regular S S -posets and CSF S S -posets. We then provide a homological classification of pomonoids based on this new property of their right (Rees factor) S S -posets.