Jtdylktuy

Let $mathfrak M = (M, 0, +, - , r)_{rin R}$ be a structure which realizes a module, i.e., $(M, 0, +, -)$ is an abelian group together with operations $r : M to M$ with $R$ a ring with $1$. Let $L_{Mod}(R) = L_{AbG} cup {r : r in R }$ with $L_{AbG} = {0,+,-}$ be the language of $R$-modules.

Definition: An equation is an $L_{Mod}(R)$-formula $gamma(overline x)$ of the form
$$
r_1 x_1 + r_2 x_2 + ldots + r_m x_m = 0.
$$
A positive primitive formula ($pp$-formula) is of the form
$$
exists overline y (gamma_1 land ldots land gamma_n)
$$
where $gamma_i(overline x, overline y)$ are equations. These definitions are taken form K.Tent/M.Ziegler, A Course in Model Theory. In these notes it is mentioned that we can combine $pp$-formulas with intersection. If $varphi(overline x)$ and $psi(overline x)$ are pp-formuals then
$$
(varphicap psi)(overline x) = varphi(overline x) land psi(overline x).
$$
But why would this give a $pp$-formula? For example if
$$
varphi(x) = exists y ( r_1 x + r_2 y = 0 ), quad
psi(x) = exists y ( s_1 x + s_2 y = 0 )
$$
then
$$
exists y ( r_1 x + r_2 y = 0 ) land
exists y ( s_1 x + s_2 y = 0 )
$$
is not a pp-formula as the existential quantifier does not appears at the beginning? Maybe we could write
$$
exists y exists z ( r_1 x + r_2 y = 0 land s_1 x + s_2 z = 0 )
$$
but this is not what is suggested in the text...

The sentence you wrote at the end is correct. Although formally, you should maybe write $$exists y, exists z, (r_1x+r_2y+0z = 0 land s_1x+0y+s_2z = 0)$$ because the definition says a pp-formula should have the form $exists z, (gamma_1(x,z)land dotsland gamma_n(x,z))$, where the $gamma_i$ are equations in the variables $xz$, i.e. they include all of the free and quantified variables.

You say that this isn't suggested in the text, but I think it is - sometimes you have to read between the lines a little!

First, you're expected to know the basic logic fact that the conjunction of two existential formulas is equivalent to an existential formula, but you have to rename any conflicting variables: $$(exists y, varphi(x,y))land (exists y, psi(x,y)) Leftrightarrow exists y, exists z, (varphi(x,y)land psi(x,z)).$$

Now the authors of the linked document write: "We can see any equation $gamma(x,y)$ in some variables as an equation $tilde{gamma}(x,y,z)$ relating more variables, simply attaching zero coefficients to the new variables. Hence we can combine two pp-formulas without mixing up existentials."

So they're suggesting the following procedure:

1. If the pp-formulas have any quantified variables in common, rename these variables in one of the formulas. This avoids "mixing up existentials".


2. Expand all equations to include all variables, by attaching zero coefficients to the new variables.


3. Take the conjunction of all of the equations, and put the existential quantifiers in front.