In lattice theory, a 0,1-simple lattice refers to a type of bounded lattice distinguished by the way it preserves its top and bottom elements under nonconstant homomorphisms. Specifically, if L is a 0,1-simple lattice and ƒ is a function from L to another lattice that preserves joins and meets—without mapping every element of L to the same value—then the function must satisfy the following property:
ƒ⁻¹(ƒ(0)) = {0} and ƒ⁻¹(ƒ(1)) = {1}.
This means that for a 0,1-simple lattice, the bottom element (0) and top element (1) are uniquely preserved by any nontrivial lattice homomorphism.
Example
Consider a flat lattice Lₙ with n atoms (a₁, a₂, …, aₙ), and with top and bottom elements denoted by 1 and 0 respectively. When n ≥ 3, the lattice Lₙ is 0,1-simple. However, when n = 2, there exists a homomorphism ƒ that maps 0 and a₁ to 0, and maps a₂ and 1 to 1, proving that L₂ is not 0,1-simple.
Applications in Lattice Theory
The concept of a 0,1-simple lattice helps mathematicians classify structures based on how their elements behave under lattice operations. It provides insights into homomorphic mappings and preservation of identity elements, contributing to deeper studies in algebraic structures, order theory, and universal algebra.
