# 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

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