The　concept　of　submodularity　finds　wide　applications　in　data　science,　artificial　intelligence,　and　machine　learning,　providing　a　boost　to　the　investigation　of　new　ideas,　innovative　techniques,　and　creative　algorithms　to　solve　different　submodular　optimization　problems　arising　from　a　diversity　of　applications.　However　pure　submodular　or　supermodular　problems　only　represent　a　small　portion　of　the　problems　we　are　facing　in　real　life　applications.　The　main　focus　of　this　work　is　to　consider　a　non-submodular　function　maximization　problem　subject　to　a　cardinality　constraint,　where　the　objective　function　is　the　sum　of　a　monotone　gamma-weakly　submodular　function　and　a　supermodular　function.　This　problem　includes　some　previously　studied　problems　as　special　cases,　such　as　the　submodular+supermodular　maximization　problem　when　gamma　=1,　and　the　gamma-weakly　submodular　function　maximization　problem　when　the　supermodular　function　is　void.　We　present　greedy　algorithms　for　this　generalized　problem　under　both　offline　and　streaming　models,　improving　existing　results.(c)　2022　Elsevier　B.V.　All　rights　reserved.