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It seems the name for this idea is equationally complete theory, see page 30 of Walter Taylor's Equational Logic survey. Not every theory is like that: for example in the theory of lattices, which is …

How about: dense linear order with endpoints. It's $\aleph_0$-categorical by the same proof as for the case without endpoints. It's not homogeneous because of the endpoints.

The Interchange lemma in the theory of formal languages has a $\Pi_5$ form: $\forall L$, if $L$ is a context-free language, then $\exists c$, $c$ is a positive integer, such that $\forall n\ge 2$, $R …

Yes. It suffices to show that any free semigroup embeds in a group. For this I refer you to MO question 3235: Let $F$ be a free semigroup (say, $2$-generated) which is embedded in a group $G$, an …

In computability theory, a set of integers $A$ is infinitely often computably traceable if there is a computable function $h$ such that for all functions $g\le_T A$ ($g$ computable from $A$), there is …

At the risk of stating the obvious, in modal logic the axiom schema $$\Box\varphi\rightarrow\varphi$$ is called the schema T.

The answer is basically "No". Consider the universe $\omega=\{0,1,2,\dots\}$ with $\in=<$. Let us define $$x\in' y\leftrightarrow x\in y\wedge \neg\exists z(x\in z\wedge z\in y).$$ Then any embedding …

It may seem a bit backwards, but one could try to look at it the other way around: pretend the axioms and theorems are the things that we observe. We don't really observe the field $\mathbb R$ of all …

Yes. If there is a model $\mathcal M$ of size $n$ of any sentence $\phi$ that does not use = then you can take any element $a$ of $\mathcal M$ (using the fact that $n>0$) and let $\mathcal N$ be $\mat …