Geodesic
Atomic theories of residuated semigroup families
Fundamentalʹnaâ i prikladnaâ matematika, Tome 6 (2000) no. 2, pp. 627-632.

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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. 
@article{FPM_2000_6_2_a19,
     author = {M. R. Pentus},
     title = {Atomic theories of residuated semigroup families},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {627--632},
     publisher = {mathdoc},
     volume = {6},
     number = {2},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2000_6_2_a19/}
}
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M. R. Pentus. Atomic theories of residuated semigroup families. Fundamentalʹnaâ i prikladnaâ matematika, Tome 6 (2000) no. 2, pp. 627-632. http://geodesic.mathdoc.fr/item/FPM_2000_6_2_a19/