TITLE: Does the following set of formulas have a finite model?
QUESTION [7 upvotes]: So this is the problem:

Do the following 3 formulas have a finite model:
  1. $\forall x \forall y (p(x,y) \Rightarrow \neg p(y,x))$
  2. $\forall x \forall y (p(x,y) \Rightarrow \exists z (p(x,z) \& p(z,y)))$
  3. $\exists x \exists y p(x,y)$

Any suggestions will be appreciated!

REPLY [3 votes]: Yes, for example, let $p \ge 4$ be a prime number congruent to $3$ mod $4$, pick $M = \Bbb Z/p\Bbb Z$ and $P(x,y) = \exists z \neq 0 (y = x+z^2)$.
Then axiom $1$ is true because $-1$ is not a square mod $p$ and axiom $2$ is true because circles have $(p+1)$ points, which is enough ($> 4$) to get nontrivial decompositions.