Associativity of two-place functions generated by left continuous monotone functions and other properties

This article introduces a weak pseudo-inverse of a monotone function, which is applied to characterize the associativity of a two-place function $T:[0,1{]}^2\to [0,1]$ defined by $T(x,y)=t^{[-1]}(F(t(x),t(y)))$ where $F:[0,\mathrm{\infty }{]}^2\to [0,\mathrm{\infty }]$ is an associative function with neutral element in $[0,\mathrm{\infty }]$, $t:[0,1]\to [0,\mathrm{\infty }]$ is a left continuous monotone function and $t^{[-1]}:[0,\mathrm{\infty }]\to [0,1]$ is the weak pseudo-inverse of $t$. It shows that the associativity of the function $T$ depends only on properties of the range of $t$. Moreover, it investigates the idempotence, the limit property, the conditional cancellation law and the continuity of the function $T$, respectively.