  Investigation of partial multiplace functions by algebraic methods plays an
important role in modern mathematics were we consider various operations on
sets of functions, which are naturally defined. The basic operation for
$n$-place functions is an $(n+1)$-ary superposition $[ ]$, but there are some
other naturally defined operations, which are also worth of consideration. In
this paper we consider binary Mann's compositions $\op{1},...,\op{n}$ for
partial $n$-place functions, which have many important applications for the
study of binary and $n$-ary operations. We present methods of representations
of such algebras by $n$-place functions and find an abstract characterization
of the set of $n$-place functions closed with respect to the set-theoretic
inclusion.
