Affine space as an algebra

Example

Let $F$ be a field and let $V$ be a vector space over $F$. The affine space of $V$, seen as an algebra in the sense of universal algebra, is the set $V$ with these operations:

1. the ternary operation $p(x, y, z) = x - y + z$;
2. for each $r \in F$, the binary operation $b_r(x, y) = (1 - r)x + ry$.

There is no constant and no addition of two points. Each operation is an affine combination, since its coefficients sum to $1$.

Check that every affine combination is a term in these operations. Take $x_1, \dots, x_n \in V$ and $\lambda_1, \dots, \lambda_n \in F$ with $\lambda_1 + \cdots + \lambda_n = 1$. Then $\sum_i \lambda_i x_i = x_1 + \sum_{i \ge 2} \lambda_i (x_i - x_1)$. Each point $b_{\lambda_i}(x_1, x_i) = x_1 + \lambda_i (x_i - x_1)$ is a term. For points $x_1 + a$ and $x_1 + c$, the term $p(x_1 + a, x_1, x_1 + c)$ equals $x_1 + a + c$. Applying $p$ repeatedly therefore gives $x_1 + \sum_{i \ge 2} \lambda_i (x_i - x_1)$. So the subalgebras are exactly the subsets closed under affine combinations, that is, the affine subspaces and the empty set.