## Notation and Terminology

#### This page describes the conventions that are used for the entries in the database

** 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 $a, b, c, d, e$.

**are denoted by lower-case roman letters, usually $x, y, z, u, v, w, x_0, x_1, \ldots$.**

*Variables that range over elements***are usually denoted by $i,j,k,m,n$.**

*Integer variables***are denoted by upper-case roman letters, usually $X, Y, Z, X_0, X_1, \ldots$**

*Variables that range over sets*
** 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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