Proper families of functions and their applications

Proper families of functions provide a convenient and memory-efficient method for specification of large parametric families of quasigroups and $d$-quasigroups of large order. We present a number of examples of proper families, outline connections between proper families and quasigroups or $d$-quasigroups, discuss equivalent definitions of properness and various operations on proper families, analyze the complexity of deciding properness and study possible configurations of essential dependence inside proper families.