partialfield

Partial field is a concept in various branches of mathematics and physics that refers to a type of mathematical structure or set of values that exhibit specific properties. In the context of group theory, a partial field is a set with two binary operations, often denoted as addition and multiplication, that satisfies certain requirements. These requirements include the existence of an additive inverse for each element, the distributive law, and the requirement that the set undergoes a change of sign under multiplication.

Similar definitions can be found in other areas of mathematics. In the study of operations on sets, a partial field often appears as a means to describe a generalized ring structure. This generalization involves relaxing certain requirements that are typically imposed on standard rings. When dealing with operations that are not necessarily globally defined on the entire set, the paradigm of a partial field serves as a framework for describing their behavior.

Partial fields can also be utilized in analyzing particular classes of mathematical expressions. In the realm of algebraic geometry, partial fields arise in studying the properties of valuations on fields. One of the chief goals in valuations theory is to describe valuation domains and spectrum of valuation rings. In such contexts, a valuation's behavior often mimics some of the characteristics of a partial field.