Question

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

Short answer

0 is the smallest element in \( \mathbb{N}^n \) under any monomial order \( \prec \).

Step-by-step solution

Understand the Problem

We need to show that the zero vector \((0, 0, \ldots, 0)\) is the smallest element with respect to a given monomial order \( \prec \) within the set \( \mathbb{N}^n \), which is the set of all n-tuples of non-negative integers.

Define Monomial Order

A monomial order \( \prec \) is a way to compare tuples in \( \mathbb{N}^n \). It is a total order on the set of monomials such that: 1) Any two elements are comparable, 2) It is compatible with multiplication by monomials, and 3) It is a well order on \( \mathbb{N}^n \).

Show 0 is in \( \mathbb{N}^n \)

The zero vector, \((0, 0, \ldots, 0)\), is clearly in \( \mathbb{N}^n \) because the set \( \mathbb{N}^n \) includes all tuples where each element is a non-negative integer.

Show Compatibility with Monomial Order

Consider any element \( \alpha = (a_1, a_2, \ldots, a_n) \) in \( \mathbb{N}^n \). By the properties of a monomial order, for any \( i \), it holds \( 0 \leq a_i \). Thus, \( (0, 0, \ldots, 0) \prec (a_1, a_2, \ldots, a_n) \).

Establish 0 as the Minimum Element

The compatibility property states that multiplying a monomial (or tuple) by a smaller or equal monomial under \( \prec \) results in another monomial that is smaller or equal under \( \prec \). Since \( (0,0,\ldots,0) \) is less than or equal to any other \( (a_1, a_2, \ldots, a_n) \) in \( \mathbb{N}^n \), it is the smallest element.

Key concepts

Zero Vector

In mathematics, the zero vector is a concept that defines a vector with all its elements as zeros. In the context of the set \( \mathbb{N}^n \), which contains all n-tuples of non-negative integers, the zero vector is written as \((0, 0, \, \ldots, \, 0)\). This vector is essential because it serves as the additive identity in vector spaces—it does not alter other vectors when added to them.
The zero vector plays a crucial role in comparing elements under a monomial order. It is important to demonstrate that this zero vector is indeed the minimal element in such ordered sets, which helps confirm its importance in algebraic structures.
When evaluating the zero vector, it's helpful to remember its inherent properties:

• All elements are zero.
• It is the smallest element in the set \( \mathbb{N}^n \) under monomial order.
• Adding it to any vector does not change that vector.

Recognizing these characteristics allows us to navigate through mathematical problems involving zero vectors with greater ease and understanding.

Natural Numbers

Natural numbers are the simplest types of numbers we encounter in math. They are

• non-negative integers
• starting from 0 and going upwards (0, 1, 2, 3, ...)

These numbers form the foundation for constructing more complex number sets.
When we talk about \( \mathbb{N}^n \), we refer to a set of n-tuples where each element is a natural number. This notation allows mathematicians to work with vectors of numbers where each component must be a natural number.
Here are some key points to understand about natural numbers and their importance:

• They are used to count objects.
• They form the basis of mathematical structures like the number line.
• They allow the definition of monomial orders on their vector iterations, as seen in \( \mathbb{N}^n \).

Thus, understanding the concept of natural numbers is crucial for grasping the construction of sets like \( \mathbb{N}^n \) and how they interact under mathematical operations.

Tuple Comparison

In mathematics, especially in algebra, you often encounter the concept of tuple comparison. This comparison is critical when establishing a monomial order over elements such as those in \( \mathbb{N}^n \). A tuple is basically an ordered list of numbers, often written as \((a_1, a_2, ..., a_n)\), and each component can be compared to corresponding components of another tuple.
To better understand tuple comparison, consider its primary function in monomial ordering:

• A total order: Each pair of tuples can be compared.
• Compatibility with multiplication: The order remains consistent even when tuples are multiplied by non-negative elements.
• Well-ordering: The set \( \mathbb{N}^n \) has a least element (the zero vector) that is smaller or equal to all other tuples.

An effective way to compare two tuples is to analyze each corresponding element sequentially, from the first through the last, until a difference is found. This comparison helps determine relationships within mathematical constructs and allows the application of rules like the monomial order. Understanding these comparisons enriches our insights into algebra and mathematical structure, and it lifts barriers to solving more involved mathematical problems.