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We introduce the concept of a Σ-bounded algebraic system and prove that if a system is Σ-bounded with respect to a subset A then in a hereditarily finite admissible set over this system there exists a universal Σ-function for the family of functions definable by Σ-formulas with parameters in A. We obtain a necessary and sufficient condition for the existence of a universal Σ-function in a hereditarily finite admissible set over a Σ-bounded algebraic system. We prove that every linear order is a Σ-bounded system and in a hereditarily finite admissible set over it there exists a universal Σ-function. Now it is generally accepted that one of the important generalizations of the concept of computability is Σ-definability (generalized computability) in admissible sets. This generalization has made it possible to study computability problems over arbitrary algebraic systems, for instance, over the field of real numbers. The most important results of computability theory in admissible sets and their applications to theoretical computer science (semantic programming, dynamic logic, the theory of effective f -spaces, and so on) are collected in the monograph of Ershov [1].One of the principal results of absolute computability theory is the existence of a universal partially computable function. It is known (see [1]) that in every admissible set there exists a universal Σ-predicate; however, this is false for Σ-functions. An algebraic system M is constructed in [2] such that in the hereditarily finite admissible set HF(M) there is no universal Σ-function. Therefore, it is interesting to know which conditions on M guarantee the existence of a universal Σ-function in the hereditarily finite admissible set HF(M) over M. It is proved in [1] that if M is an algebraic system of a decidable and model complete theory then in HF(M) there exists a universal Σ-function. For a certain class K of algebraic systems, necessary and sufficient conditions for the existence of a universal Σ-function in HF(M), where M ∈ K, were found in [3][4][5]. A torsion-free abelian group A was constructed in [6] such that in HF(A) there is no universal Σ-function.In this article we introduce the concept of a Σ-bounded (with respect to a finite subset) algebraic system. We prove that if M is a Σ-bounded system with respect to a finite subset M 0 then in HF(M) there exists a universal Σ-function for the family of all Σ-functions determined by Σ-formulas with parameter M 0 . We obtain a necessary and sufficient condition for the existence of a universal Σ-function in the hereditarily finite admissible set over a Σ-bounded algebraic system. We prove that every linear order is a Σ-bounded system and in the hereditarily finite admissible sets over it there exist universal Σ-functions. As regards terminology and notation for admissible sets, we follow [1]. Also, we use the following notation. Take an algebraic system M of a finite signature σ 0 , whose underlying set is denoted by M , and some subset M 0 ⊆ M . Put σ 1 = {σ 0 , ∈, U 1 , ∅} and σ i (M 0 ) = σ i ∪...

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