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Difference between revisions of "Clone"

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m (tex encoded by computer)
m (fixing dots)
 
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$$  
 
$$  
 
\omega _ {n}  ^ {i}
 
\omega _ {n}  ^ {i}
( a _ {1} \dots a _ {i} \dots a _ {n} )  =  a _ {i}  $$
+
( a _ {1}, \dots, a _ {i}, \dots, a _ {n} )  =  a _ {i}  $$
  
for any  $  n $-
+
for any  $  n $-tuple  $  ( a _ {1}, \dots, a _ {n} ) $
tuple  $  ( a _ {1} \dots a _ {n} ) $
 
 
in  $  A  ^ {n} $,  
 
in  $  A  ^ {n} $,  
 
where  $  n \geq  1 $,  
 
where  $  n \geq  1 $,  
$  i = 1 , 2 \dots $
+
$  i = 1 , 2, \dots $
 
and  $  A $
 
and  $  A $
is an arbitrary fixed set. By a composition of operations  $  \omega _ {1} ( x _ {1} \dots x _ {j} \dots x _ {n} ) $
+
is an arbitrary fixed set. By a composition of operations  $  \omega _ {1} ( x _ {1}, \dots, x _ {j}, \dots, x _ {n} ) $
and  $  \omega _ {2} ( y _ {1} \dots y _ {m} ) $
+
and  $  \omega _ {2} ( y _ {1}, \dots, y _ {m} ) $
one means the operation  $  \omega _ {3} ( z _ {1} \dots z _ {l} ) $
+
one means the operation  $  \omega _ {3} ( z _ {1}, \dots, z _ {l} ) $
 
defined by the formula
 
defined by the formula
  
 
$$  
 
$$  
\omega _ {1} ( x _ {1} \dots x _ {j-} 1 ,\  
+
\omega _ {1} ( x _ {1}, \dots, x _ {j-1} ,\  
\omega _ {2} ( y _ {1} \dots y _ {m} ) ,\  
+
\omega _ {2} ( y _ {1}, \dots, y _ {m} ) ,\  
x _ {j+} 1 \dots x _ {n} )
+
x _ {j+1}, \dots, x _ {n} )
 
$$
 
$$
  
 
for some  $  j \leq  n $,  
 
for some  $  j \leq  n $,  
where the sets of variables  $  X = \{ x _ {1} \dots x _ {j} \dots x _ {n} \} $,  
+
where the sets of variables  $  X = \{ x _ {1}, \dots, x _ {j}, \dots, x _ {n} \} $,  
$  Y = \{ y _ {1} \dots y _ {m} \} $
+
$  Y = \{ y _ {1}, \dots, y _ {m} \} $
and  $  Z = \{ z _ {1} \dots z _ {l} \} $
+
and  $  Z = \{ z _ {1}, \dots, z _ {l} \} $
 
satisfy the equality
 
satisfy the equality
  

Latest revision as of 06:39, 22 February 2022


of an operation

Any set of finitary operations of the form $ \omega : A ^ {n} \rightarrow A $ that is closed with respect to composition and contains all the projections $ \omega _ {n} ^ {i} : A ^ {n} \rightarrow A $, defined by

$$ \omega _ {n} ^ {i} ( a _ {1}, \dots, a _ {i}, \dots, a _ {n} ) = a _ {i} $$

for any $ n $-tuple $ ( a _ {1}, \dots, a _ {n} ) $ in $ A ^ {n} $, where $ n \geq 1 $, $ i = 1 , 2, \dots $ and $ A $ is an arbitrary fixed set. By a composition of operations $ \omega _ {1} ( x _ {1}, \dots, x _ {j}, \dots, x _ {n} ) $ and $ \omega _ {2} ( y _ {1}, \dots, y _ {m} ) $ one means the operation $ \omega _ {3} ( z _ {1}, \dots, z _ {l} ) $ defined by the formula

$$ \omega _ {1} ( x _ {1}, \dots, x _ {j-1} ,\ \omega _ {2} ( y _ {1}, \dots, y _ {m} ) ,\ x _ {j+1}, \dots, x _ {n} ) $$

for some $ j \leq n $, where the sets of variables $ X = \{ x _ {1}, \dots, x _ {j}, \dots, x _ {n} \} $, $ Y = \{ y _ {1}, \dots, y _ {m} \} $ and $ Z = \{ z _ {1}, \dots, z _ {l} \} $ satisfy the equality

$$ Z = ( X \setminus \{ x _ {j} \} ) \cup Y ,\ \ m , l \geq 1. $$

Comments

The name "clone" was invented by P. Hall. It first appeared in print in (the first edition of) [a1].

References

[a1] P.M. Cohn, "Universal algebra" , Reidel (1981)
How to Cite This Entry:
Clone. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Clone&oldid=46362
This article was adapted from an original article by V.B. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article