Binomial Ideals and Congruences on $$\mathbb {N}^n$$

Abstract

A congruence on N n is an equivalence relation on N n that is compatible with the additive structure. If k is a field, and I is a binomial ideal in k[X 1 , . . . , X n ] (that is, an ideal generated by polynomials with at most two terms), then I induces a congruence on N n by declaring u and v to be equivalent if there is a linear combination with nonzero coefficients of X u and X v that belongs to I. While every congruence on N n arises this way, this is not a oneto-one correspondence, as many binomial ideals may induce the same congruence. Nevertheless, the link between a binomial ideal and its corresponding congruence is strong, and one may think of congruences as the underlying combinatorial structures of binomial ideals. In the current literature, the theories of binomial ideals and congruences on N n are developed separately. The aim of this survey paper is to provide a detailed parallel exposition, that provides algebraic intuition for the combinatorial analysis of congruences. For the elaboration of this survey paper, we followed mainly [10] with an eye on [5] and [13].

Numerical Semigroups, Polyhedra, and Posets III: Minimal Presentations and Face Dimension

“…The kernel of ϕ S is the equivalence relation ∼ = ker ϕ S that sets z ∼ z whenever ϕ S (z) = ϕ S (z ). Each such relation z ∼ z is called a trade of S. The kernel ∼ is in fact a congruence, meaning that z ∼ z implies (z + z ) ∼ (z + z ) for all z, z , z ∈ N k+1 ; see [17,21] for a survey on the role of congruences in this context. A subset ρ ⊂ ker ϕ S , viewed as a subset of N k+1 × N k+1 , is a presentation of S if ker ϕ S is the smallest congruence on N k+1 containing ρ; see [22,Propsition 8.4] for a thorough description of the smallest congruence containing a given set of relations.…”

Minimal Systems of Binomial Generators for the Ideals of Certain Monomial Curves

“…. , x n and t are indeterminates, is a binomial ideal (see [1], or [2] for a more recent reference). Clearly, ker(ϕ A ) is the defining ideal of a monomial curve.…”