Question

Show that \(0=(0, \ldots, 0) \in \mathbb{N}^{n}\) is the smallest element with respect to a monomial order \(\prec\) : \(0=\min _{\prec} \mathbb{N}^{n}\).

Short answer

The zero vector \((0, 0, ..., 0)\) is the smallest in any monomial order on \(\mathbb{N}^n\).

Step-by-step solution

Understanding the Problem

We need to show that the zero vector \(0 = (0, 0, ewline . . . , 0)\) is the smallest element in \(\mathbb{N}^n\) concerning any given monomial order \(\prec\). This means that, under the order \(\prec\), no other vector \(a = (a_1, a_2, . . . , a_n)\) in \(\mathbb{N}^n\) is smaller than \((0, 0, ewline . . . , 0)\).

Properties of Monomial Orders

Monomial orders are total orders on the set of monomials that are compatible with multiplication and have the property that \(1\) (equivalently \((0, 0,ewline . . . , 0)\)) is the smallest element. This means in any monomial ordering, the origin or zero vector must necessarily be the least.

Showing \((0, 0, ..., 0)\) is the Smallest

Since monomial orders dictate that the zero vector \((0, 0, ewline . . . , 0)\) must be less than or equal to any other vector \((a_1, a_2, ewline . . . , a_n)\) in \(\mathbb{N}^n\), we have \( (0,0,ewline . . . , 0) \prec (a_1, a_2, ewline . . . , a_n)\). This illustrates that no other monomial, no matter the order, can be smaller.

Conclusion about Monomial Order

By the definition of a monomial order, the zero vector \((0,0,ewline . . . ,0)\) is necessarily the smallest element in \(\mathbb{N}^n\), thus proving \(0 = \min_{\prec} \mathbb{N}^n\). This conclusion holds regardless of the specific monomial order chosen.

Key concepts

Zero Vector

In mathematics, particularly in the context of vectors, the zero vector, which is represented as \(0 = (0, 0, \ldots, 0)\), is a fundamental concept. This vector is composed entirely of zeros across its coordinates. The zero vector is unique in that it is not just zero in magnitude,but it also functions as a neutral element in vector addition. When you add the zero vector to any other vector, the original vector remains unchanged.

• Example: For any vector \( a = (a_1, a_2, \ldots, a_n) \), we have \( a + 0 = a \).
• It’s the foundational vector for defining directions and dimensions. It sets the origin in geometric space.

In the context of monomial orders within \( \mathbb{N}^n \) (where \( \mathbb{N}^n \) represents n-tuples of natural numbers),the zero vector represents the smallest element under any monomial order. This is because no other vector with all its components beingnatural numbers can have a value less than zero for each coordinate. The monomial order thus ensures that the zero vectorholds a sort of foundational or base status in ordered sets of vectors.

Total Order

A total order is a crucial concept in mathematics when comparing and arranging elements within a set. It provides a full ranking system, where every element can be compared to every other element in a specified comprehensive manner. This means that for any two elements in a set, one element is either smaller, larger, or equivalent to the other.

• This ordering system is transitive. If element a is related to b, and b is related to c, then a is related to c.
• It's reflexive, meaning every element is comparable to itself.
• It’s antisymmetric so that if one element is equivalent to another, then they must be the same element in terms of value.

For monomial orders, which is a type of total order, the significance lies in its ability to compare all monomials or vectors in \( \mathbb{N}^n \) systematically. This ensures a hierarchy that defines one monomial as smaller than, equally, or greater than another.In our context of monomial orders, this property formalizes the assertion that the zero vector, standing as the smallest monomial, is consistent across all ordering.

Natural Numbers

Natural numbers are the set of positive integers starting from 0 or 1, depending on context, and proceeding indefinitely upward. They are denoted by the symbol \( \mathbb{N} \) and are foundational in mathematicsfor counting and ordering.

• They include numbers like 0, 1, 2, 3, and so forth.
• These numbers are inherently greater than zero except for the zero itself.
• Each number has a successive number obtained by adding one.

In the context of the zero vector and \( \mathbb{N}^n \), this means we're dealing with vectors where each coordinate must be anatural number.This stipulation is vital because it ensures that in any comparison or ordering of vectors, the consideration starts with zero as the potential smallest value.Since all numbers in \( \mathbb{N} \) are non-negative, when used in a vector form, they reinforce the zero vector’s uniqueness as the minimal element by virtue that all other natural numbercomponents can only be positive or zero.