Singular points in generalized concatenation structures that otherwise are homogeneous

Abstract

A generalized concatenation structure X is any ordered relational structure in which at least one relation is a function of two or more variables that is monotonically increasing in each of them. A point of X is said to be singular if it is fixed under all automorphisms of the structure. Examples are 0 in the additive real numbers, the velocity of light in special relativity, and the status quo in some theories of utility. A translation is an automorphism with no fixed points other than the singular ones. It is assumed that X is order dense and finitely unique, from which it follows (Theorem 1) that at most three singular points may exist; a maximum, a minimum, and/or an interior one. Assuming also that X is homogeneous in translations between adjacent singularities, the singular points are further characterized (Theorem 2), with the interior one being shown to be a generalized zero (Definition 4). It is shown how to replace a generalized zero by a zero (Theorem 3) and how to partition X into homogeneous structures on either side of the interior generalized zero (Theorem 4). The major problem in representing such structures lies in understanding how the two halves relate. For structures on a continuum that are translation homogeneous between singularities, finitely unique, and solvable relative to the zero, a representation exists in terms of unit structures (Theorem 5).