|WikiProject Mathematics||(Rated C-class, Low-priority)|
|A summary of this article appears in Subalgebra#Subalgebras in universal algebra.|
|A summary of this article appears in Embedding#Universal algebra and model theory.|
Rescued comment from the talk page of submodel
Substructure and submodel are different enough ideas in model theory that conflating them will mislead people about key ideas of the field, particularly the meaning of "theory" and "language". Take the theory of fields in the language of rings, for example (with symbols for 1, 0, binary addition, unary negation, and binary multiplication). A field may have subrings that are not subfields (the rationals have the integers as a subring) - these are substructures without being submodels.
The conditions to make a subtructure are weaker than those of a submodel, because substructures can violate existential axioms of the theory: substructures can be found inside models by just specifing some of the elements the structure contains, and then closing under whatever functions are in the language - but they aren't necessarily models of the given theory themselves, because they could be missing elements required by the theory, but not supplied by functions in the language (again, consider the ring of integers within the field of rationals). If you want all your substructures to be submodels, you have to tune the language for that purpose.Isotropy 22:44, 30 July 2006
- I agree, although in practice I have never seen any confusion caused by this. I think these subtle points are best explained by covering both notions within one article. Therefore I am merging the "submodel" stub into "substructure" stub and extending both.--Hans Adler (talk) 19:09, 25 November 2007 (UTC)
Scope of this article
- Not in this article. This is an article about mathematics. There is a very clear pointer to the article superstructure for the discussion of substructures in an engineering context. Hans Adler 17:26, 15 January 2010 (UTC)
In the submodel section, it is stated that "...the even numbers (2Z, +, 0) form a submodel which is (a group but) not a subgroup...".
Shouldn't the even numbers form a subgroup of the integers?