Upper and Lower Approximations of Subsemimodules in Semimodules over Semirings

This paper develops a rough set–theoretic framework for semimodules over commutative semirings using the Bourne relation induced by subsemimodules and partitioning subsemimodules. Upper and lower approximations of subsets and subsemimodules are introduced and systematically analyzed. Necessary and sufficient conditions under which these approximations constitute subsemimodules are established, with particular emphasis on subtractive extensions. Several examples are provided to illustrate the theory. The results extend rough set theory to semimodule contexts and furnish a coherent algebraic framework for additive inverse-free structures.