Multiplicative lattices with absorbing factorization
Andreas Reinhart, Gulsen Ulucak
TL;DR
This work generalizes the notion of $1$-absorbing prime ideals to the setting of $C$-lattices by introducing $O$A-elements (the $1$-absorbing prime elements) and studying their factorization behavior. It develops two parallel factorization theories—OA-factorization (OAFLs) and TA-factorization (TAFLs)—and their variants for compact and principal elements (CTAFLs/PTAFLs and COAFLs/POAFLs), establishing precise dimensional and structural constraints, localization results, and interconnections with classical lattice classes such as $ZPI$-lattices and Prüfer lattice domains. Key contributions include characterizations of when OA-elements coincide with primes in quasi-local and principallgenerated lattices, dimension bounds for TAFLs, and several equivalences linking OAFLs/TAFLs with $ZPI$- and Prüfer properties in various lattice contexts (e.g., quasi-local, Prüfer lattice domains). These results advance multiplicative lattice theory by tying absorbing-type factorizations to well-known lattice classes and localization phenomena, with implications for factorization theory in the lattice setting.
Abstract
In [24], Yassine et al. introduced the notion of 1-absorbing prime ideals in commutative rings with nonzero identity. In this article, we examine the concept of 1-absorbing prime elements in C-lattices. We investigate the C-lattices in which every element is a finite product of 1-absorbing prime elements (we denote them as OAFLs for short). Moreover, we study C-lattices having 2-absorbing factorization (we denote them as TAFLs for short).