  Tropical mathematics often is defined over an ordered cancellative monoid
$\tM$, usually taken to be $(\RR, +)$ or $(\QQ, +)$. Although a rich theory has
arisen from this viewpoint, cf. [L1], idempotent semirings possess a restricted
algebraic structure theory, and also do not reflect certain valuation-theoretic
properties, thereby forcing researchers to rely often on combinatoric
techniques.
  In this paper we describe an alternative structure, more compatible with
valuation theory, studied by the authors over the past few years, that permits
fuller use of algebraic theory especially in understanding the underlying
tropical geometry. The idempotent max-plus algebra $A$ of an ordered monoid
$\tM$ is replaced by $R: = L\times \tM$, where $L$ is a given indexing semiring
(not necessarily with 0). In this case we say $R$ layered by $L$. When $L$ is
trivial, i.e, $L=\{1\}$, $R$ is the usual bipotent max-plus algebra. When
$L=\{1,\infty\}$ we recover the "standard" supertropical structure with its
"ghost" layer. When $L = \NN $ we can describe multiple roots of polynomials
via a "layering function" $s: R \to L$. Likewise, one can define the layering
$s: R^{(n)} \to L^{(n)}$ componentwise; vectors $v_1, \dots, v_m$ are called
tropically dependent if each component of some nontrivial linear combination
$\sum \a_i v_i$ is a ghost, for "tangible" $\a_i \in R$. Then an $n\times n$
matrix has tropically dependent rows iff its permanent is a ghost.
  We explain how supertropical algebras, and more generally layered algebras,
provide a robust algebraic foundation for tropical linear algebra, in which
many classical tools are available. In the process, we provide some new results
concerning the rank of d-independent sets (such as the fact that they are
semi-additive),put them in the context of supertropical bilinear forms, and lay
the matrix theory in the framework of identities of semirings.
