On the Category of Quantale-Semimodules

The concepts from quantale theory is applied in this work to bring forward and study an abstract idea of quantale-semimodule (Q-semimodule) and Q-sets. We have introduced the category \(\varvec{Q}\)-Mod of Q-semimodule and confer an adjunction between the category \(\varvec{Q}\)-Mod and the well-known category Set, and another between the category \(\varvec{L}\)-Set of L-sets and the category \(\varvec{Q}\)-Set of Q-sets. Finally, we have shown that the category \(\varvec{Q}\)-Mod forms a monoidal category.