Clone

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[edit]

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

#### References[edit]

[a1] |  P.M. Cohn, "Universal algebra" , Reidel (1981)

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