Tensor algebra, which includes vector-like operations such as addition and scalar multiplication, defines key products like the outer product, which creates a higher-rank tensor, and the contracted product, a generalization of the inner product formed by contraction, an operation that reduces rank by two. A tensor's inherent properties, like symmetry or anti-symmetry, allow for a unique decomposition of any rank-two tensor, while the Quotient Law is a crucial theorem used to verify if a mathematical object behaves as a true tensor.
