Atomic theories of residuated semigroup families

A residuated semigroup is a partially ordered semigroup together with two binary operations $\backslash$ and $/$, such that the assertions $a\leq c/b$, $a\cdot b\leq c$, and $b\leq a\backslash c$ are equivalent. We formulate a necessary and sufficient condition for an arbitrary set of atomic formulas of the signature $\{\leq,\cdot,\backslash,/\}$ to be the atomic theory of some residuated semigroup family. We also consider some specific residuated semigroups and residuated semigroup families.