signs to separate each member in the set. If some of the simplexes have the opposite orientation, these are prefixed by a minus sign. If some of the simplexes occur in the set more than once, these are prefixed with an integer count. Thus, an affine chain takes the symbolic form of a sum with integer coefficients. Note that each facet of an n-simplex is an affine (n − 1)-simplex, and thus the boundary of an n-simplex is an affine (n − 1)-chain. Thus, if we denote one positively oriented affine simplex as with the denoting the vertices, then the boundary of σ is the chain It follows from this expression, and the linearity of the boundary operator, that the boundary of the boundary of a simplex is zero: Likewise, the boundary of the boundary of a chain is zero: . More generally, a simplex (and a chain) can be embedded into a manifold by means of smooth, differentiable map . In this case, both the summation convention for denoting the set, and the boundary operation commute with the embedding. That is, where the are the integers denoting orientation and multiplicity. For the boundary operator , one has: where ρ is a chain. The boundary operation commutes with the mapping because, in the end, the chain is defined as a set and little more, and the set operation always commutes with the map operation (by definition of a map). A continuous map to a topological space X is frequently referred to as a singular n-simplex. (A map is generally called "singular" if it fails to have some desirable property such as continuity and, in this case, the term is meant to reflect to the fact that the continuous map need not be an embedding.) Algebraic geometry Since classical algebraic geometry allows one to talk about polynomial equations but not inequalities, the algebraic standard n-simplex is commonly defined as the subset of affine (n + 1)-dimensional space, where all coordinates sum up to 1 (thus leaving out the inequality part). The algebraic description of this set is which equals the scheme-theoretic description with the ring of regular functions on the algebraic n-simplex (for any ring ). By using the same definitions as for the classical n-simplex, the n-simplices for different dimensions n assemble into one simplicial object, while the rings assemble into one cosimplicial object (in the category of schemes resp.