6
votes
1answer
106 views

Using the compactness theorem to disprove axiomatizability

Another model-theoretic exercise from Smirnov's book. Problem: Construct infinite family of varieties such that their union is not axiomatizable. My solution: Denote by $\mathcal{A}_n$ the variety ...
3
votes
1answer
39 views

Elementary equivalence of models

I'm quite new to model theory, so please correct me if I'm using wrong terminology. I need help with an exercise from Smirnov's book "Varieties of algebras" (In Russian). Problem: Assume that a ...
1
vote
1answer
61 views

Are algebraic structures required to satisfy axioms?

Is it a requirement for algebraic structures, when studying universal algebra, to satisfy axioms? The reason I ask is because algebraic structures are only defined by a underlying set, a signature, ...
3
votes
2answers
124 views

Should axioms be viewed as part of the signature?

I included category theory in the tags in order to get feedback from the categorial logic community. It goes without saying that this isn't really category theory. A semigroup can be defined as a ...
4
votes
1answer
47 views

Which sentences survive the passage from $X$ to the set of all functions $I \rightarrow X$?

Suppose $X$ is a mathematical structure with a single underlying set which we will also denote $X$, equipped with some functions and relations. Letting $I$ denote an arbitrary non-empty set, we see ...
0
votes
0answers
54 views

Medium-strong (graph) homomorphisms

Weak (graph) homomorphisms are mappings $f: V(G) \rightarrow V(G')$ such that the images of connected nodes $x,y$ (in the source graph) are connected: $$R(x,y) \rightarrow R(f(x),f(y)) = R(x',y')$$ ...
1
vote
1answer
48 views

Existence of arbitrarily large ordinal subgroups in a group structure on a regular cardinal [duplicate]

Suppose $\kappa$ is an uncountable regular cardinal, and $(\kappa, \cdot, ^{-1}, e$) is a group. Prove that that $C = \{\alpha < \kappa: \alpha\, \textrm{is a subgroup of}\, \kappa)$ is unbounded ...
1
vote
1answer
95 views

Idea for a proof involving an identity of term functions on $\sigma$-structures [x]

I have some problems with the following theorem: Fix an signature $\sigma$ and a set of variables $\mathbb{V}$. We call $t$ and $t_1$ "equivalent", if for every $\sigma$-structure $S$ and every term ...