Talk:Structure (mathematical logic)

message: Structure actually has two uses in math. logic: this and a fairly well established usage in structural proof theory due to Belnap.  Shall we move this to Structure (model theory), and create Structure (proof theory)? --- Charles Stewart(talk) 19:40, 15 February 2006 (UTC)
replies: text: How common is the latter usage, really? My gut reaction is to leave this article where it is, and put a dab line at the top to the proof-theory concept, but I won't argue about it if you think the two notions have comparable footprints and want to have a dab page. --Trovatore 20:22, 15 February 2006 (UTC)
thread_title: Ambiguity

message: Some examples would be useful in this article (eg some simple examples on the integers with the natural ordering; eg a simple Hasse diagram of a poset). reetep 12:36, 17 November 2006 (UTC)
thread_title: More Examples Needed

message: "In database theory, structures with no functions are studied as models for relational databases, in the idiosyncratic form of relational models."
Is there a rationale or citation for relational models being described as "idiosyncratic"? And is this referring to The Relational Model, or to relational models of particular datasets?
I guess I'm also a little puzzled as to the point of mentioning databases at all. Sure, both Structure(math) and Relational Model identify a thing of interest, and a collection of features of that thing, and can be operated upon with logic of some type, but is there much more to it than that?  What about semantic nets?  Or programming language type systems?  Gwideman (talk) 23:09, 10 December 2009 (UTC)
replies: text: The point is that there is significant overlap between a part of database theory and a part of finite model theory. They research exactly the same things, but one side calls the objects of study structures, and the other calls them databases. As happens often in computer science, the abstract model for databases is extremely clumsy from a mathematical point of view. It is close to implementation in a computer and almost impossible to reason about precisely. I think I used the word "idiosyncratic" a long time ago, with no source. Nowadays I am a bit more careful about "original research", and it doesn't express exactly what I wanted to say anyway. I will just remove it. Hans Adler 01:27, 11 December 2009 (UTC)
thread_title:  Database comments in this article 

message: In the "Definition" section, the impression is given that a structure is only defined by its signature and not by axiomatic relations such as the existence of identity elements or inverses.  But then, in the section "Induced substructures and closed subsets", it is mentioned that the rationals are the smallest substructure of the reals, generated by the empty set.  This is not consistent.  If only the <+,*,-,0,1> signature is relevant, then the integers Z are a smaller substructure.  If on the other hand the axiomatic relations are also important, the "Definition" section needs to be more explicit on this.  —Preceding unsigned comment added by 217.111.146.68 (talk) 08:58, 22 June 2010 (UTC)
replies: text: This is a decent point, although I'm not sure in either case that it's exactly axioms that we're talking about.  Rather, my immediate reaction is that the signature should also include the multiplicative inverse function (or, equivalently, division).  I imagine the only reason whoever wrote the section didn't include it is that the zero of the field doesn't have an inverse.  That is a slight annoyance, and I'm not sure how best to handle it in the text. --Trovatore (talk) 09:09, 22 June 2010 (UTC)
replies: text: The signature of fields doesn't include a multiplicative inverse function, though, so I'm going to fix the article. — Carl (CBM · talk) 10:58, 22 June 2010 (UTC)
replies: text: Is there really a standard "signature of fields"?  Clearly there's more than one choice you could use.  --Trovatore (talk) 17:58, 22 June 2010 (UTC)
replies: text: I've seen some slight variants (e.g. whether − is unary or binary) but never one that contained division or inverse. Algebraist 18:05, 22 June 2010 (UTC)

text: In model theory, they usually take the signature of fields to be the signature of rings, which is to say they study the model theory of fields in the signature of rings. E.g. Hodges' Shorter Model Theory p. 10, Marker's Model Theory p. 17. — Carl (CBM · talk) 18:16, 22 June 2010 (UTC)
replies: text: In that case maybe fields aren't a very good example for this article.  --Trovatore (talk) 18:22, 22 June 2010 (UTC)
replies: text: I think the point that a substructure doesn't have to satisfy the same theory as the original structure is somewhat relevant (cf. section 1.4). I think there is a nice parallelism between the second and third paragraphs of section 2.1. — Carl (CBM · talk) 18:27, 22 June 2010 (UTC)
thread_title: Definition

message: The definition quotes:
:\mathcal A is called an (induced) substructure of \mathcal B if
:...
:* the interpretations of all function and relation symbols agree on |\mathcal B|.
Shouldn't this read:
:* the interpretations of all function and relation symbols agree on |\mathcal A|.
since |\mathcal A|\subseteq |\mathcal B|? — Ron van den Burg (talk) 13:29, 27 February 2011 (UTC)
thread_title:  Typo in: Induced substructures and closed subsets ? 

message: The article says:
:Given two structures \mathcal A and \mathcal B of the same signature σ, a (σ-)homomorphism from \mathcal A to \mathcal B is a map h:|\mathcal A|\rightarrow|\mathcal B| which preserves the functions and relations. More precisely:
:...
:* For every n-ary relation symbol R of σ and any elements a_1,a_2,\dots,a_n\in|\mathcal A|, the following implication holds:
:::(a_1,a_2,\dots,a_n)\in R \implies (h(a_1),h(a_2),\dots,h(a_n))\in R.
However, since the interpretations of symbol R (R^{\mathcal A} and R^{\mathcal B}) are different, shouldn't the line read:
:* For every n-ary relation symbol R of σ and any elements a_1,a_2,\dots,a_n\in|\mathcal A|, the following implication holds:
:::(a_1,a_2,\dots,a_n)\in R^{\mathcal A} \implies (h(a_1),h(a_2),\dots,h(a_n))\in R^{\mathcal B} .
instead? — Ron van den Burg (talk) 13:50, 27 February 2011 (UTC)
replies: text: Yes this looks correct, i.e. I agree with you. 69.143.122.185 (talk) 23:54, 26 January 2023 (UTC)

text: I just edited the article to make the necessary changes (the conflation of the relation symbol in the object theory with its interpretations in the two structures A and B occurs at least twice in that section and I tried to fix all occurrences). 69.143.122.185 (talk) 00:07, 27 January 2023 (UTC)
thread_title:  Homomorphisms: shouldn't the distinction between the symbol R and its interpretations be relevant here? 

message: Just to elaborate a little more on my undo in this edit.
It is true that at some level you can always replace a function by a relation giving the same information.  For a unary function symbol f, you can substitute a relation symbol R, and interpret it via R(x,y) if and only if f(x)=y.
However, there are significant differences between languages that contain function symbols and ones that have only relation symbols.  Function symbols have existential import, in the sense that if you have a structure for a language with a function symbol f, and a closed term t in the language, then the structure must contain an interpretation for f(t).  Relation symbols don't give you that; a structure for a language containing our relation symbol R above is not guaranteed to have an object y such that the structure satisfies R(t,y).
This is important for lots of tools used to analyze structures.  For example, the Ehrenfeucht–Fraïssé game is usually defined for languages that have only relation and constant symbols.  If you allow non-nullary function symbols, the analysis gets more complicated. --Trovatore (talk) 22:17, 5 April 2015 (UTC)
replies: text: The current Wikipedia defintion of the Ehrenfeucht–Fraïssé game is dissonant with your argument, as it specifically excludes function symbols:  "Suppose that we are given two structures \mathfrak{A} and \mathfrak{B}, each with no function symbols and the same set of relation symbols..."  How would you suggest this be resolved? Jim Bowery (talk) 06:42, 7 April 2015 (UTC)
replies: text: Well, I was being conservative.  I think there's a way to make them work with function symbols, but I don't really know how to do it.  Since I have a vague notion that there's some way to get around the problem with sufficient work, I didn't want to flat-out assert that the technique can't be used at all. --Trovatore (talk) 20:59, 7 April 2015 (UTC)
thread_title:  Explanation of undo 

message: Seems like as soon as we discuss the operations on structures, strangely, set theory pops up and everything is a set now. That's a little bit strange, because then these notions are not applicable to set theory, which is not based on sets; it's also not applicable to lambda-calculus, which has nothing to do with sets.
Can we get rid of using the term "set" at this level? It sounds like it's all turtles all the way down, with turtles replaced with sets.
Vlad Patryshev (talk) 02:44, 30 August 2021 (UTC)
replies: text: Tarski's model-theoretic semantics for first-order logic uses a set theory (usually ZF or ZFC) to implement semantics for first-order logic theories. I.e. the use of sets is inherent to the field of model theory. So I don't understand the objection here. 69.143.122.185 (talk) 23:53, 26 January 2023 (UTC)
thread_title:  Why Sets 

message: Satisfaction relation seems to have a typo:
It has both \mathcal {M} and M. M seems to be a typo for \mathcal {M}.
If it isn't a typo, where M has appeared from? This needs to be explicitly said. --77.127.25.154 (talk) 08:26, 14 December 2021 (UTC)
replies: text: . The definition in section Structure_(mathematical_logic)#Definition says "\mathcal A=(A, \sigma, I)". This was tacitly applied to the letter "M", assuming tacitly "\mathcal{M} = (M, \sigma, I)" (I have made this explicit). To make the confusion complete, "M" was written as "M" (I have fixed this). - Thanks for noticing! - Jochen Burghardt (talk) 09:03, 14 December 2021 (UTC)
thread_title:  Math font typo in "Satisfaction relation"? 

message: The definition used in this article for "strong homomorphism" seems strange on the surface because it a priori assumes that the homomorphism is surjective. The problem with this definition has been discussed on math.stackexchange https://math.stackexchange.com/questions/466657/two-definitions-of-strong-homomorphism, where they concluded that the definition in the Wikipedia article was possibly incorrectly copied from Chang, Keisler, which apparently assumes all homomorphisms are surjective throughout the book.
Another problem with the definition given in this article is the claim (which is in this article) that elementary embeddings are exactly the injective strong homomorphisms appears to be false. This is also mentioned in the math.stackexchange post. The claim doesn't appear to be true using this definition of "strong homomorphism". (Trying to better understand elementary embeddings is actually how I ended up in this section of the article.)
The definition of "strong homomorphism" that is ultimately agreed upon the math.stackexchange post is (i) f is a homomorphism (so preserves relations) and (ii) f also reflects relations i.e. for all \bar{a} one has  R^B(f(\bar{a})) \implies R^A (\bar{a}) . The same definition is also used here https://pdaniell.farm/archives/strong-homomorphisms as well as in this other math.stackexchange post https://math.stackexchange.com/questions/1988514/what-is-strong-homomorphism?noredirect=1&lq=1
I would edit the article myself to get rid of the current definition and use the definition of "strong homomorphism" found on all of these websites, but I do not know (yet) of a textbook using this definition. I guess if after searching through model theory textbooks I find one that defines the concept I can edit the article later to at least mention that there is another definition out there. 69.143.122.185 (talk) 23:51, 26 January 2023 (UTC)
thread_title:  Strong homomorphism definition