Dedekindchains

Dedekindchains are a concept in abstract algebra, specifically within the study of ordered structures. They are named after the mathematician Richard Dedekind. A Dedekindchain is a finite sequence of elements in a partially ordered set, say x_0, x_1, ..., x_n, such that for each i from 0 to n-1, the element x_i is covered by x_{i+1}. The notion of "covered" is crucial here. In a partially ordered set, an element 'a' is said to cover an element 'b' if b < a and there is no element 'c' such that b < c < a. This means that 'a' is the immediate successor of 'b' in the ordering.

Dedekindchains are often used to describe the structure of certain lattices and other ordered algebraic objects. The length of a Dedekindchain can provide information about the "height" or complexity of the ordered set. For example, in the context of Boolean algebras, the length of the longest Dedekindchain corresponds to the rank of the algebra. The study of these chains helps in understanding the relationships between elements and the overall organization of the poset. They are a fundamental tool for analyzing the local structure of ordered sets.