Math Problem Statement
( R , = ) is this poset a lattice?
Solution
To determine whether the poset is a lattice, we must understand the definition of a lattice and check if satisfies it.
Lattice Definition
A poset is a lattice if for every pair of elements , there exist:
- A least upper bound (also known as the supremum or join), denoted by , which is the smallest element in greater than or equal to both and .
- A greatest lower bound (also known as the infimum or meet), denoted by , which is the largest element in less than or equal to both and .
Poset
In the poset :
- The set represents the set of all real numbers.
- The relation is the usual order on the real numbers.
Analyzing the Poset
For any two real numbers :
- The least upper bound (supremum) of and exists and is given by .
- The greatest lower bound (infimum) of and exists and is given by .
Since both the supremum and infimum exist for every pair of elements in , the poset is indeed a lattice.
Conclusion
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Math Problem Analysis
Mathematical Concepts
Posets
Lattices
Formulas
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Theorems
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Suitable Grade Level
Advanced Mathematics