Commutative formal group law

One-dimensional formal group law
Let $$R$$ be a commutative unital ring. A one-dimensional formal group law on $$R$$ is a formal power series $$F$$ in two variables, denoted $$x$$ and $$y$$, such that:

Note that conditions (1)-(3) alone define formal group law (which is not necessarily commutative). Also, condition (3) is redundant.

Two examples of commutative formal group laws, both of which work for any commutative unital ring, are the additive formal group law and the multiplicative formal group law.

Higher-dimensional formal group law
Let $$R$$ be a commutative unital ring. A $$n$$-dimensional formal group law is a collection of $$n$$ formal power series $$F_i$$ involving $$2n$$ variables $$(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n)$$ satisfying a bunch of conditions.

Before stating the conditions, we introduce some shorthand. Consider $$x = (x_1,x_2,\dots,x_n)$$ and $$y = (y_1,y_2,\dots,y_n)$$. Then, $$F(x,y)$$ is the $$n$$-tuple $$(F_1(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n),F_2(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n),dots,F_n(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n))$$.

Note that conditions (1)-(3) define a (not necessarily commutative) formal group law. Condition (3) is redundant, i.e., it can be deduced from (1) and (2).