# Definition:Big Monoid Ring

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## Definition

Let $\struct {M, \cdot}$ be a divisor-finite monoid.

Let $\struct {R, +, \times}$ be an additive semiring.

The **big monoid ring** of $R$ over $M$ is the ringoid $\struct {R^M, +, *}$ where:

- $R^M$ be the set of all mappings $M \to R$
- $+$ is the pointwise operation induced by $+$
- $*$ denotes convolution of mappings

## Also known as

The **big monoid ring** is also known as the **total algebra** of $M$ over $R$.

## Also see