Augmented Algebra

An augmentation of an Associative Unital Algebra $A$ over a Semiring $R$ is an $R$ Algebra Homomorphism $\varepsilon:A\to R$, where $R$ is viewed as an associative unital algebra over itself.

This lets us view the algebra $A$ as a structure that is parameterized by $R$. In particular, the $(R,R)$-bimodule structure allows us to reparameterize an element $A$ via actions of $r:R$.

Examples

• The Tensor Algebra of an $(R,R)$ Semiring Bimodule $M$ is comes equipped with an augmentation $\varepsilon:T^{n}(M)\to R$ defined as

  $\par\varepsilon(\overline{m})=\begin{cases}1&m=1\\ 0&\text{otherwise}\end{cases}\par$