$\Sigma$-Bounded algebraic systems and universal functions. I

We introduce the concept of a $\Sigma$-bounded algebraic system and prove that if a system is $\Sigma$- bounded with respect to a subset $A$ then in a hereditarily finite admissible set over this system there exists a universal $\Sigma$-function for the family of functions definable by $\Sigma$-formulas with parameters in $A$. We obtain a necessary and sufficient condition for the existence of a universal $\Sigma$-function in a hereditarily finite admissible set over a $\Sigma$-bounded algebraic system. We prove that every linear order is a $\Sigma$-bounded system and in a hereditarily finite admissible set over it there exists a universal $\Sigma$-function.