Notation and terminology
From Structures
This page describes the conventions that are used for the entries in the database
Use the 'Edit' link to clarify, modify or add further conventions.
Sets are denoted by upper-case roman letters, usually $A, B, C,\ldots, U, V, W$.
$\mathbb{N}=$ the set of natural numbers $=\{0,1,2,\ldots\}$,
$\mathbb{Z}=$ the set of integers $=\mathbb{N}\cup\{-n:n\in\mathbb{N}\}$,
$\mathbb{Q}=$ the set of rationals $=\{m/n:m,n\in\mathbb{Z}, n>0\}$,
$\mathbb{R}=$ the set of real numbers,
$\mathbb{C}=$ the set of complex numbers $=\{x+iy:x,y\in\mathbb{R}\}$.
$\mathcal P(A)=\{S:S\subseteq A\}$, the power set of $A$.
$A^n=\{\langle a_0,\ldots,a_{n-1}\rangle:a_0,\ldots,a_{n-1}\in A\}$, the set of all $n$-tuples of elements of $A$.
Elements of sets are denoted by lower-case roman letters, usually $a, b, c, d, e$.
Variables that range over elements are denoted by lower-case roman letters, usually $x, y, z, u, v, w, x_0, x_1, \ldots$.
Integer variables are usually denoted by $i,j,k,m,n$.
Variables that range over sets are denoted by upper-case roman letters, usually $X, Y, Z, X_0, X_1, \ldots$
Functions are denoted by lower-case roman letters, usually $f, g, h$.
A (first-order) operation on a set $A$ is a function from $A^n$ to $A$, where $n\ge 0$ is the arity of the operation. If $n=0$ then the operation is called a constant.
A (first-order) relation on a set $A$ is a subset of $A^n$, where $n>0$ is the arity of the relation.
A second-order operation on a set $A$ is a function from $\mathcal P(A)^n$ to $A$.
A second-order relation on a set $A$ is a subset of $\mathcal P(A)^n$.
A mathematical structure is a tuple of the form $\mathbf{A}=\langle A,\ldots\rangle$ where $A$ is a set and $\ldots$ specifies a list of (possibly higher-order) operations and relations on $A$.
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