notation  meaning  basis  Alternative 

direct sum or Kronecker sum depending on context  
direct product or Kronecker product depending on context  
/\  exterior algebra  
/\^{k}  exterior algebra k^{th} blade  
scalar or real depending on context (see below)  e0  
²  2 dimensional vector  e1,e2  
³  3 dimensional vector  e1,e2,e3  
^{n}  n dimensional vector  e1…en  
^²²  bivector based on 2D algebra  e12  
^²³  bivector based on 3D algebra  e12,e31,e23  
C  complex numbers  
H  quaternions  ³  
O  octonions  ^{7}  
aA  a is an element of the set A  
I have used the term 'scalar' interchangeably with the term 'real', that is, a continuous value that can be represented by a single number.
Strictly speaking the term 'scalar' should be reserved for a quantity that is used to scale a vector, that is change its magnitude without changing its direction, or in other words a scalar is the ratio of parallel vectors.
For instance I should not really call energy a scalar because there are no vectors involved.
I apologise for my lack of mathematical rigor here, its just that the word scalar seems to better express that it is not a vector and its less likely to cause confusion with the real part of a complex number. Also this (mis?)usage is quite common in the computer world.
notation  meaning 

[M]  matrix 
[M]^{t}  transpose of matrix (swap rows & columns) 
[M]^{1}  inverse of matrix 
[I]  identity matrix (ones on leading diagonal, otherwise zeros) 
The individual elements of the matrix are numbered as follows,
m_{00}  m_{01}  m_{02}  …  m_{0n} 
m_{10}  m_{11}  m_{12}  …  m_{1n} 
m_{20}  m_{21}  m_{22}  …  m_{2n} 
m_{p0}  m_{p1}  m_{p2}  …  m_{pn} 
The first subscript represents the row, the second subscript represents the column
notation  example  meaning  pages in site 

subscript  e_{1}  coordinate basis  
superscript  x^{1}  coordinate value  
Einstein Summation Convention  e_{i}x^{i}=e_{1}x^{1}+e_{2}x^{2}+e_{3}x^{3}=∑e_{i}x^{i}  When the same index appears twice in an expression, once raised and once lowered, a sum is implied.  
partial derivatives  ∂_{a}=∂/∂^{a} 
notation  example  meaning  

A  A = {a  aA}  set  
a  aA  element of A  
{a, b, c}  set containing a, b and c  
a H = {ah  hH}  left coset of a  
(a,b)  cycle notation  
<a,b>  all the elements of group generated by a and b  
<r,f  r³=1, f²=1, frf=r^{1}>  generators with constrains  

K = {k  kK}  complement of set indicated by bar over top  
 A   grade (size) = number of elements in A  
 G : H   number of left cosets of H in G  
×  Direct Product  
Semidirect Product (left)  
{n, h}_{φ} {n',h'}  Semidirect Product (right)  
Bicrossed Product also known as Knit or ZappaSzep product 

h = h'H such that a h' = h a 
Normal subgroup a H = H a for all a in G  
Ø  empty set  
¬  not  
∧  and (= intersection)  
∨  or (U = union)  
is in (is an element of the set)  
AB  A is contained in B (or A is a subset of B)  
→  if  
<>  if and only if  
for all  
there exists  
is isomorphic to  
(S)  The set of subsets of S (power set)  
orb(s)  Orbit  set of elements that can be reached  
stab(s)  Stabilizer  set of elements that don't move s 
Sets are discussed on this page and groups here.
Endo is a Greek word meaning inside.
notation  example  meaning  pages in site 

/\  conjunction, and , intersection (in set), greatest lower bound (in order) 

\/  disjunction, or , union (in set), least upper bound (in order)  
T  Top (true)  
__  Bottom (false)  
=>  A => B  Implication A => B means A implies B 

Tφ  φ is a theorem of T In contrast to 'implication' above this is a statement about statements usually written only when we mean φ follows from T, wheras A => B is a statement we may consider as a conjecture . See [Lawvere,Rosebrugh page 198] Not to be confused with right adjoint in category theory. 
logic  
for all (quantifier)  
there exists (quantifier) 
notation  example  meaning  pages in site 

AB  Map / Morphism / Function between mathematical structures that relates or preserves the structures in some way. May be injection, surjection or bijection (injection can be specifically indicated see below)  
a b  A function in terms of elements of the structures aA, bB  
A function that is valid if all the functions drawn as solid lines are valid  
injective morphism (embedding)  
canonical map  map of G onto factor group G/H where H is a normal subgroup of G  
Left adjoint  
Right adjoint 
notation  example  meaning  pages in site 

[[D]]  [[2*5]] = 10  A 'denotion' is written in double square brackets. It represents the meaning of D to a program evaluation of D 

Bottom (undefined)  
Less defined than 
upper  lower  upper  lower  upper  lower  

Alpha  A  α  Iota  I  ι  Rho  P  ρ 
Beta  B  β  Kappa  K  κ  Sigma  Σ  σ 
Gamma  Γ  γ  Lambda  Λ  λ  Tau  Τ  τ 
Delta  Δ  δ  Mu  M  μ  Upsilon  Υ  υ 
Epsilon  E  ε  Nu  N  ν  Phi  Φ  φ 
Zeta  Z  ζ  Xi  Ξ  ξ  Chi  X  χ 
Eta  H  η  Omicron  O  ο  Psi  Ψ  ψ 
Theta  Θ  θ  Pi  Π  π  Omega  Ω  ω 
×  ÷  ±  •  
°  ð  Ð  ƒ  µ  
Ø  ø  Þ  þ  †  
…  ∂  ◊  ≤  
≥  ∈  ≠  ≈  ∫  
∞  ∀  ∃  √  
m  10^{3}  K  10^{3} 
µ  10^{6}  M  10^{6} 
n  10^{9}  G  10^{9} 
p  10^{12}  T  10^{12} 
f  10^{15}  
a  10^{18} 
metadata block 

see also: 
external 
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