On a generalization of I-regularity

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>S</m:mi> </m:math> S be a pomonoid. The projectivity and strong flatness of right <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>S</m:mi> </m:math> S -posets have been central topics in the homological classification of pomonoids in recent decades. In 2005, Shi et al. introduced <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>I</m:mi> </m:math> I -regular <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>S</m:mi> </m:math> S -posets and proved that all its cyclic <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>S</m:mi> </m:math> S -subposets are projective [ Indecomposable, projective, and flat S-posets , Comm. Algebra 33 (2005), no. 1, 235–251]. Feng subsequently introduced CSF <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>S</m:mi> </m:math> S -posets, where all its cyclic <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>S</m:mi> </m:math> S -subposets are strongly flatness [CSF <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>S</m:mi> </m:math> S -posets and C(P) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>S</m:mi> </m:math> S -posets, Master’s thesis, Northwest Normal University, Gansu, 2020]. In this article, we define a new class of strongly (SF)-cyclic <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>S</m:mi> </m:math> S -posets, which lies strictly between the classes of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>I</m:mi> </m:math> I -regular <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>S</m:mi> </m:math> S -posets and CSF <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>S</m:mi> </m:math> S -posets. We then provide a homological classification of pomonoids based on this new property of their right (Rees factor) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>S</m:mi> </m:math> S -posets.