Shiftomorphisms and Binary Operations

By Walker Carlisle

A binary operation is a mathematical function that takes two things and yields a single new thing. One common binary operation is addition, which adds two numbers to give a single new number, the sum. Both addition and another binary operation, multiplication, can be defined on the same set of numbers, which can thus create what is called a field; for example, the real numbers are said to be “a field under addition and multiplication.” I introduce the notion of three or more binary operations defined on the same set of numbers. I specifically include traditional addition and multiplication and introduce new yet analogous binary operations that behave similarly but have new and interesting properties. Thus, by analogy, a field-like mathematical structure is created. If two fields behave in such a similar way that they can be thought of as the same field under two different names, then they are said to be “isomorphic,” and the function relating them to each other is an “isomorphism.” The most rigorous approach to defining a set with three binary operations seems to require the notion of one field nested inside another isomorphic one in such a way that addition in the smaller field behaves exactly as multiplication in the larger field. The term “shiftomorphism” refers to an isomorphism obeying these properties. I explore the behavior of this structure and use it to obtain results regarding many binary operations on one set. The results could have applications in real and complex analysis and field theory.