A glossary of terms

This web page describes various terms used in these logic web pages.

aka: Abbreviation for also known as.
abelian group: A group for which the binary operation is commutative. Also called a commutative group.
bijection: A function $f : A \to B$ which is both an injection and a surjection. Bijections are also called one-to-one correspondences.
canonical: Meaning according to a set of rules, or canon. This word is used in mathematics to mean that the definition of the object in question is forced upon us in some way: either because it is the simplest or most natural such definition that works or more usually because it is the only such definition.
embedding: A map or function taking a structure $A$ (such as a group, ring, field, etc.) into another similar structure $B$ , so that the image of $A$ (considered as a substructure of $B$ ) looks exactly the same as $A$ . Such a function will always be injective and preserve any binary operations present.
group: A set with a binary operation that is associative and has identity and inverses.
identification: When two structures look identical (such as a structure and the image of it via an embedding) it often makes sense to regard the two structures as really being the same. We say that we identify them. Such identifications are not strictly logically correct, because the two copies of the same object really are different copies, but the simplification gained is always worthwhile. Examples include identifying the integers with the copy of the integers in an ordered field.
injection: A function $f : A \to B$ such that $f (x) \neq f (y)$ for all $x \neq y$ from $A$ . Injections are also called one-to-one functions.
isomorphism: A map or function taking a structure $A$ (such as a group, ring, field, etc.) exactly onto another similar structure $B$ , so that both $A$ (considered as a substructure of $B$ ) and $B$ look exactly the same. In other words, an isomorphism is an embedding that is surjective as well as injective.
surjection: A function $f : A \to B$ such that for all $y \in B$ there is $x \in A$ with $y = f (x)$ . Surjections are also called onto functions.