3
votes
1answer
61 views

Best algebra text for Model Theory

I'm looking for an algebra book that is tailored towards some of the ideas in Model Theory, I'm currently slogging through Hodges' Model Theory. I'm a bit rusty with my algebra and was curious if ...
3
votes
1answer
59 views

real closure of an archimedean field

my question is: Is an archimedean field dense in its real closure? I know that in the non-archimedean case, this does not have to be true (e.g., rational fucntions). Thanks!
4
votes
1answer
37 views

When a subgroup of automorphism group of a structure is in the form of automorphism group of a substructure?

Question 1: Is the following statement true? ($*$) Let $\mathcal{L}$ be a first order language and $\mathcal{M}$ a $\mathcal{L}$-structure and $H\leq Aut(\mathcal{M})$ then there exists a ...
2
votes
0answers
39 views

Counter example of monotone union [duplicate]

I saw this exercise in "Elements of Abstract and Linear Algebra" by E. H. Connell: Suppose $G$ is a group. Suppose $T$ is an index set and for each $t \in T$, $H_t$ is a subgroup of $G$. Furthermore, ...
11
votes
2answers
231 views

Putting down axioms for some symbols. Playing with their consequences qualitatively and symbolically. Building theories. The book?

I am interested in the design and building of theories. By building theories, I mean putting down axioms of various kinds, over various fields, exploring their perhaps interesting, or probably boring, ...
2
votes
0answers
51 views

The composition of functors and algebraic structures

An algebraic structure can be viewed as a finite-product-preserving functor $X : \mathbf{L} \rightarrow \mathbf{C},$ where $\mathbf{L}$ is a Lawvere theory and $\mathbf{C}$ is a category. Thus given ...
2
votes
1answer
57 views

Problem concerning formally real fields

I'm trying to reconcile a fact I am reading in David Marker's Model Theory text. He claims on page 326 that $\mathbb{F}=\mathbb{Q}(\sqrt{2}, \sqrt{-2})$ is a formally real field. This seems like it ...
1
vote
1answer
46 views

Introduction to Valued Fields

I'm looking for an introductory text on valued fields, to be used as the basis for a reading group for model theorists. Currently, I know of one such text, Valued Fields by Engler/Prestel. However, ...
2
votes
3answers
100 views

Prove $\forall r \in \mathbb{R}. \exists k \in \mathbb{Z}. r < k$

I would like to prove that for every real number there exists an integer that is greater than it. My problem lies in that I am not sure how to construct the real numbers and provide their theory with ...
1
vote
1answer
103 views

Renaming the elements of a mathematical structure

One of the most basic insights about mathematical structures is that we can rename their elements without fundamentally changing the structure. Question. How do we actually formalize this observation ...
7
votes
5answers
406 views

What is exactly the meaning of being isomorphic?

I know that the concept of being isomorphic depends on the category we are working in. So specifically when we are building a theory, like when we define the natural numbers, or the real numbers, or ...
4
votes
1answer
67 views

Class of finite groups a Fraïssé Class? [duplicate]

Is the class of finite groups a Fraïssé class? Calling this class $K$, does $K$ satisfy the following: Joint embedding property Amalgamation property Hereditary property: if $G \in K$ and $H \le G$, ...
0
votes
2answers
39 views

cardinality of elements in a “semiring minus multiplicative identity”

In a theory that has all axioms of semiring except multiplicative identity axiom, will there be a model of the theory that has infinite elements? The model must violate multiplicative identity axiom.
0
votes
2answers
83 views

how do we prove that ring of characteristic $p$ has arbitrarily large models?

As title says, how do we prove that the theory that describes ring of characteristic $p$ has arbitrarily large model? I am asking for a model-theoretic approach.
3
votes
2answers
96 views

Non-Archimedan Groups

I'm trying to think of an explicit example of a non-Archimedian group. The definition of Archimiedean is s.t. if for all $x$ and $y$, there is some $m$ a Natural number s.t. $mx = \underbrace {x + x ...
5
votes
3answers
214 views

How can we tell if a set of axioms uniquely determines an algebraic structure?

Up to isomorphism. For instance, the group axioms are verified by an infinite number of non-isomorphic algebraic structures. But the Peano axioms, I think (my proof may lack some formality due to my ...
4
votes
1answer
273 views

why algebraic structures?

According to wikipedia, an algebraic structure is an arbitrary set with one or more finitary operations defined on it. From a model theory perspective, I understand this definition as: structure with ...
2
votes
1answer
105 views

Understanding the model-theoretic proof of Hilbert's Nullstellensatz

The proof I am talking about goes like this: Given $k$ algebraically closed and $(f_1,..,f_k)=I\neq (1)$ an ideal in $A=k[x_1,..,x_n]$, let $m$ be a maximal ideal with $I\subseteq m$ and observe that ...
2
votes
0answers
47 views

Model complete theories of henselian local rings which are not nec valuation rings

I just want to ask if anybody as any examples of a first order model complete theorie of henselian local rings which is not some theory of valuation rings. More precisely- I am looking for a theory ...
1
vote
1answer
145 views

Is $\mathbb N$ definable in $\mathbb C$?

$\mathbb C$ is an algebraic closed field with characteristic $0$, hence $Th(\mathbb C)$ is a recursive satisfiable complete theory, thus recursive axiomatizable. So if $\mathbb N$ is definable in ...
2
votes
2answers
49 views

Definable orders

Let $(K, <)$ be an order field, can I define the order "<" in $K$ ? I know that $K \models 0<a \;$ if and only if there is $b$ in the real closure of $K$ such that $b*b = a$. Can I ...
1
vote
0answers
51 views

Density for archimidean extension of real closed field

Let $(k,<)$ be a real closed field and $L|K$ an ordered extension such that $\forall x\in L \exists y\in k\; (x<y)$. Is $k$ dense in $L$?
6
votes
2answers
123 views

What's the name of this operator?

Let $f,g$ be functions in $C^A$ and $C^B$ respectively. Let $\boxtimes:C^A \times C^B \to (C\times C)^{A \times B}$ s.t. $f\boxtimes g(a,b)=(f(a),g(b))$ It seems not the tensor product, nor ...
0
votes
1answer
193 views

Non-triviality of a semigroup and a semiring

Assume we are given an additive semigroup $M$ which we know it is non-trivial i.e. $M\neq \lbrace 0 \rbrace$. Let $R$ be the semiring obtained from adding a multiplication law to the semigroup. Under ...