Graded (mathematics) | Wikipedia audio article
Channel: wikipedia tts
Date: 07/10/2019

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SUMMARY
=======
In mathematics, the term “graded” has a number of meanings, mostly related:
In abstract algebra, it refers to a family of concepts:

An algebraic structure 
  
    
      
        X
      
    
    {\displaystyle X}
   is said to be 
  
    
      
        I
      
    
    {\displaystyle I}
  -graded for an index set 
  
    
      
        I
      
    
    {\displaystyle I}
   if it has a gradation or grading, i.e. a decomposition into a direct sum 
  
    
      
        X
        =
        
          ⨁
          
            i
            ∈
            I
          
        
        
          X
          
            i
          
        
      
    
    {\displaystyle X=\bigoplus _{i\in I}X_{i}}
   of structures; the elements of 
  
    
      
        
          X
          
            i
          
        
      
    
    {\displaystyle X_{i}}
   are said to be “homogeneous of degree i”.
The index set 
  
    
      
        I
      
    
    {\displaystyle I}
   is most commonly 
  
    
      
        
          N
        
      
    
    {\displaystyle \mathbb {N} }
   or 
  
    
      
        
          Z
        
      
    
    {\displaystyle \mathbb {Z} }
  , and may be required to have extra structure depending on the type of 
  
    
      
        X
      
    
    {\displaystyle X}
  .
Grading by 
  
    
      
        
          
            Z
          
          
            2
          
        
      
    
    {\displaystyle \mathbb {Z} _{2}}
   (i.e. 
  
    
      
        
          Z
        
        
          /
        
        2
        
          Z
        
      
    
    {\displaystyle \mathbb {Z} /2\mathbb {Z} }
  ) is also important.
The trivial (
  
    
      
        
          Z
        
      
    
    {\displaystyle \mathbb {Z} }
  - or 
  
    
      
        
          N
        
      
    
    {\displaystyle \mathbb {N} }
  -) gradation has 
  
    
      
        
          X
          
            0
          
        
        =
        X
        ,
        
          X
          
            i
          
        
        =
        0
      
    
    {\displaystyle X_{0}=X,X_{i}=0}
   for 
  
    
      
        i
        ≠
        0
      
    
    {\displaystyle i\neq 0}
   and a suitable trivial structure 
  
    
      
        0
      
    
    {\displaystyle 0}
  .
An algebraic structure is said to be doubly graded if the index set is a direct product of sets; the pairs may be called “bidegrees” (e.g. see spectral sequence).
A 
  
    
      
        I
      
    
    {\displaystyle I}
  -graded vector space or graded linear space is thus a vector space with a decomposition into a direct sum 
  
    
      
        V
        =
        
          ⨁
          
            i
            ∈
            I
          
        
        
          V
          
            i
          
        
      
    
    {\displaystyle V=\bigoplus _{i\in I}V_{i}}
   of spaces.
A graded linear map is a map between graded vector spaces respecting their gradations.
A graded ring is a ring that is a direct sum of abelian groups 
  
    
      
        
          R
          
            i
          
        
      
    
    {\displaystyle R_{i}}
   such that 
  
    
      
        
          R
          
            i
          
        
        
          R
          
            j
          
        
        ⊆
        
          R
          
            i
            +
            j
          
        
      
    
    {\displaystyle R_{i}R_{j}\subseteq R_{i+j}}
  , with 
  
    
      
        i
      
    
    {\displaystyle i}
   taken from some monoid, usually 
  
    
      
        
          N
        
   ...