Elliptic loops

Given a local ring $(R,\mathfrak{m})$ and an elliptic curve $E(R/\mathfrak{m})$, we define loops as the points of $\mathbb{P}^2(R)$ projecting to $E$ under canonical modulo-$\mathfrak{m}$ reduction, endowed with operation that extends curve's addition. While their subset satisfying Weierstrass equation is group, these larger objects are proved be power associative abelian algebraic loops, which seldom completely associative. When loop has no order $3$, its affine part obtained stratification one-parameter family curves defined over $R$, call layers. Stronger associativity properties established when $\mathfrak{m}^e$ vanishes for small values $e \in \mathbb{Z}$. underlying $R = \mathbb{Z}/p^e\mathbb{Z}$, infinity generated by two elements, group structure layers may same projection possess geometric description.