Essentially Finite Chains

摘要

A sequence whose terms begin with a specific term (the originator) and whose subsequent terms are determined by a specific iteration procedure (the forumulator) such that each inherits the structure of its precessor, is called a chain. Examples are given to clarify the meaning of 'essentially.' The paper examines a number of instances in which an essentially infinite chain becomes essentially finite, and gives the necessary conditions on the dimensionality of the originator for the chains to be essentially finite and specifies the member at which the chain terminates. Where G is a group and F is the process of selecting any proper subgroup of G, the modification to F seems largely independent of the nature of G, except that G should have finite dimension, which permits explication in a setting of finite abelian groups and their endomorphisms. (Author)