The meaning of integral semigroup algebra

Question

Let $B$ be any Boolean algebra. I saw the term the integral semigroup algebra $\mathbb{Z}[B]$. What is the meaning of the integral semigroup algebra $\mathbb{Z}[B]$?

Answer 1

If you have any commutative semigroup $(S, \cdot)$ (written with multiplicative notation), and any commutative ring $R$, the semigroup $R$-algebra $R[S]$ is a new commutative ring, defined like this $$R[S]= \left\{ \sum_{i=0}^N a_i s_i : a_i \in R, s_i \in S \right\}$$ these are thought as finite formal sums. Sum is componentwise, while product is defined using distributivity: $$ \left( \sum_{i} a_i s_i \right) \cdot \left( \sum_{j} b_j x_j \right) = \sum_{i,j} (a_i b_j) s_i \cdot x_j $$

In practice, this is the $R$-algebra formed by adjoining to $R$ the elements of the semigroup $S$, preserving the structure of the semigroup (it's not the 'free' $R$-algebra).

The adjective integral refers to the ring of integers $R= \Bbb Z$. You can find these definitions and further details at this link.