1 vector space aka module

• Module R M
  • [Scalar semi-ring R] - \(S(+,\cdot)\) typical addition and multiplication
    • represents scalar multiplication of vectors
  • [additive communitive monoid M] - basically a monoid that is also communitive
    • represents linear combination of vectors

A module is a generalization of vector spaces to a scalar semiring

2 Subspace aka submodule

• Subspace R M
  • division ring R
  • additive communitative group M

3 Tropical algebra

• remember semi-ring \(S(+,\cdot)\) is just typical add,multiply
  
• replace that with \(S(min,+)\)