Some new results on quasi-ordered residuated systems

Quasi-ordered residuated system is a commutative residuated integral monoid ordered under a quasi-order was introduced in 2018 by Bonzio and Chajda as a generalization of commutative residuated lattices and hoop-algebras. This paper introduces the concept of atoms in these systems and analyzes its properties. Additionally, two extensions of the system $\mathfrak{A}$ to the system $\mathfrak{A}\cup\{a\}$ were designed so that the element $w$ is an atom in $\mathfrak{A}\cup\{a\}$.