Connections with Model Theory
Elementary Equivalence and Elementary Substructures
Structures indistinguishable by first-order sentences, and substructures that agree with the ambient structure on every formula.
Learning objectives
- Define elementary equivalence and elementary substructure
- Distinguish both from isomorphism and from ordinary substructure
- Give examples separating the notions
Elementary equivalence
Structures A and B for the same language are elementarily equivalent, written A ≡ B, if they satisfy exactly the same first-order sentences.
Isomorphic structures are elementarily equivalent, since an isomorphism preserves satisfaction. The converse fails dramatically.
| Pair | Why not isomorphic |
|---|---|
| R and a non-standard real closed field | Different cardinality or non-Archimedean |
| N and a non-standard model of arithmetic | The non-standard model has elements above every numeral |
| Q and R as dense linear orders without endpoints | Different cardinality; the theory is complete so they are equivalent |
| An infinite algebra and its ultrapower | Different cardinality in general |
By Löwenheim–Skolem, any theory with an infinite model has models of every infinite cardinality above the language size. So elementary equivalence can never pin down an infinite structure up to isomorphism.
Elementary substructures
A substructure agrees on atomic formulas. An elementary substructure agrees on all formulas, including those with quantifiers — so existential claims witnessed in B must be witnessed inside A.
A substructure that is not elementary
The even integers form a substructure of the integers under addition. But ∃y (y + y ≈ x) holds in the integers for x = 2 with witness 1, and fails in the even integers, where 1 is absent. So the inclusion is not elementary.
The three relations compared
- Isomorphic — same up to relabelling
- Elementary substructure — contained and agrees on all formulas
- Elementarily equivalent — same first-order theory
- Substructure — contained, agrees on atomic formulas only
- Elementarily equivalent — same first-order theory
- Elementary substructure — contained and agrees on all formulas
The vertical relationships are not a single chain: elementary substructure implies both elementary equivalence and substructure, but the latter two are independent of each other.
| ⇒ Elementarily equivalent | ⇒ Substructure | |
|---|---|---|
| Isomorphic | Yes | No — different universes |
| Elementary substructure | Yes | Yes |
| Substructure | No | Yes |
| Elementarily equivalent | Yes | No |
Why the notions matter for algebra
- Elementary equivalence bounds what identities can express. Two elementarily equivalent algebras satisfy the same identities, so identities cannot distinguish them.
- Ultrapowers are elementarily equivalent to their base. This is a corollary of Łoś's theorem and is the standard source of non-isomorphic elementarily equivalent pairs.
- Elementary substructures preserve algebraic properties. Simplicity, subdirect irreducibility and other first-order-expressible conditions transfer along elementary inclusions.
- Not everything is first-order. Being finitely generated, being simple in the presence of infinitely many congruences, and being free are not generally first-order properties, so they need not transfer.
Frequently asked questions
Can a proper elementary substructure be the same size as the whole structure?
Yes, for infinite structures. The rationals with order have proper elementary substructures of the same cardinality, obtained by removing suitable subsets.
Is every substructure of a finite structure elementary?
No. The same failure as the even integers can occur in the finite case whenever an existential witness lies outside the substructure.
Source. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, The Millennium Edition — a corrected re-typesetting of Springer-Verlag Graduate Texts in Mathematics 78 (1981). Section V.1, book pages 226-230.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates and reorganises mathematical results; it is not a reproduction of the source text.