Carroll's Theorem

Carroll's Theorem states that in certain mathematical structures called *lattices,* if you know the rules for combining elements (called joins and meets), then you can uniquely determine the order or hierarchy among those elements. Essentially, it means that the way elements relate to each other can be fully understood from how they combine, allowing you to reconstruct the order using only the algebraic operations. This theorem helps mathematicians understand complex systems where the relationships are defined by how parts combine, such as in logic, set theory, and computer science.