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# Defining algebraic elements

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This paper is a survey and a synthesis of the various approaches to defining algebraic elements. Most of it is devoted to proving the following result.

Let T be a universal theory with the Amalgamation Property. Then for T the notions of algebraic element introduced by Robinson, Jónsson, and Morley are identical. Furthermore, they extend in a natural way the notion of algebraic element introduced by Park, and used by Lachlan and Baldwin and by Kueker.

In the course of proving this we shall construct the algebraic closure as a suitable injective hull and prove a unique factorisation theorem for algebraic predicates.

We shall also show (in §3) that if T is closed under products then algebraic elements all have degree 1. Thus in algebra, algebraic elements reduce to epimorphisms.

To demonstrate the remarkable stability of the notion we shall show (at the end of §5) that defining algebraic elements by infinitary formulas yields no new ones.

Let L be a language. The cardinality of the set of formulas of L is denoted by ∣L∣. An L-theory is a deductively closed set of L-sentences. We let denote the category of models of T and (L-structure) homomorphisms between them. If A is a substructure of B we write AB. We call u: AB an injection if u is an isomorphism of A with a substructure of B, and let denote the subcategory of consisting of all injections.

## References

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[1]Bacsich, P., An epi-reflector for universal theories, Canadian Mathematical Bulletin (to appear).
[2]Bacsich, P., Model theory of epimorphisms, Canadian Mathematical Bulletin (to appear).
[3]Baldwin, J. and Lachlan, A., On strongly minimal sets, this Journal, vol. 36 (1971), pp. 7996.
[4]Banaschewski, B., Injectivity and essential extensions in equational classes of algebras, Proceedings of the Conference on Universal Algebra, Queen's papers, no. 25.
[S]Eklof, P. and Sabbagh, G., Model-completions and modules, Annals of Mathematical Logic, vol. 3 (1971), pp. 251295.
[6]Grätzer, G., Universal algebra, Van Nostrand, Princeton, N.J., 1968.
[7]Jónsson, B., Algebraic extensions of relational systems, Mathematica Scandinavica, vol. 11 (1962), pp. 179205.
[8]Jónsson, B., Extensions of relational structures, The theory of models, North-Holland, Amsterdam, 1965.
[9]Kueker, D., Generalised interpolation and definability, Annals of Mathematical Logic, vol. 1 (1970), pp. 423468.
[10]Morley, M., Categoricity in power, Transactions of the American Mathematical Society, vol. 114 (1965), pp. 514538.
[11]Park, D., Set theoretic constructions in model theory, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, Mass., 1964.
[12]Robinson, A., On the metamathematics of algebra, North-Holland, Amsterdam, 1951.
[13]Robinson, A., Introduction to model theory and the metamathematics of algebra, North-Holland, Amsterdam, 1965.
[14]Robinson, A., On the notion of algebraic closedness for noncommutative groups and fields, this Journal, vol. 36 (1971), pp. 441444.
[15]Sacks, G., Saturated model theory, Benjamin, New York, 1972.

# Defining algebraic elements

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