Theorems about Systems: Wedderburnian Theorems

A system, to exist, has to have at least one component.

A system may be single-component or multi-component.

(Note that ‘single-component’ means that more than one component is not allowed, while ‘multi-component’ does not necessarily mean that there is more than one component, but simply that more than one component is allowed.)

Note that for single-component systems there is no distinction between global and local, and so, canonically, neither term is used in describing theorems about such systems.

A system, or a certain component of the system, may or may not be lacking in development / growth. For example, in a division ring, multiplication is underdeveloped, in that it is not guaranteed to be commutative. A field, on the other hand, is not lacking such development.

A Wedderburnian theorem is a theorem of the form ‘P implies Q’ about a system, such that Q describes a higher development / growth of the system as a whole or one of its components. Wedderburnian theorems are of three types, two about multi-component systems and one about single-component systems. We will call them, respectively, type 1, type 2, and type 3, as given in the definitions below.

Definition 1 A Wedderburnian theorem of type 1 is a Wedderburnian theorem of the form ‘globally infinite and locally finite implies local development’, about a multi-component system.

An example of a Wedderburnian theorem of type 1 is Kőnig’s lemma (a special case of which is the tree-forking theorem: Every infinite tree having the finite-forking property has an infinite branch.)

Definition 2. A Wedderburnian theorem of type 2 is a Wedderburnian theorem of the form ‘globally finite implies local development’, about a multi-component system.

Two examples of a Wedderburnian theorem of type 2 are Folkman’s theorem, and Croot’s theorem (re: the Erdős-Graham problem).

Definition 3. A Wedderburnian theorem of type 3 is a Wedderburnian theorem of the form ‘finite implies further development’, about a single-component system,

An example of a Wedderburnian theorem of type 3 is Wedderburn’s theorem (Every finite division ring is a field.)

Definition 4. ‘nonce’ for a multi-component system means ‘local’, and for a single-component system is a vacuous placeholder term.

Thus, all Wedderburnian theorems have as their conclusion ‘nonce development’.

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