I have been thinking about the concept of specialization of algebras defined over the field of rational functions $k = \mathbb{C}(t)$ (I'm using $\mathbb{C}$ for my work, but the question can be asked over any field).
In my examples the algebras are quadratic, but I don't think that this is essential.
The main question I want to ask is this: suppose I have a PBW-algebra over $k = \mathbb{C}(t)$ that is specializable in an appropriate sense (described below).
When can I conclude that the specialized algebra has the same size as the original?

Setup
-----

The setup uses the language of reduction systems as in George Bergman's (excellent) paper The Diamond Lemma for Ring Theory.
Start with a finite set $X = \{ x\_1, \dots, x\_n \}$ of generators, let $\langle X \rangle$ denote the free semigroup with unit on $X$ (i.e. the words on the alphabet $X$), and let $k \langle X \rangle$ be the free algebra on $X$.

A reduction system is a set $S$ of pairs $\sigma = (W\_\sigma, f\_\sigma)$, where $W\_\sigma \in \langle X \rangle$ and $f\_\sigma \in k \langle X \rangle$.
Then the associated $k$-algebra is
$$
A = k \langle X \rangle / (W\_\sigma - f\_\sigma \mid \sigma \in S).
$$
Now we assume that we have a semigroup partial order $\leq$ on $\langle X \rangle$ (with descending chain condition), compatible with multiplication in the sense that $ A \leq A'$ implies that $BAC \leq BA'C$ for all $B,C \in \langle X \rangle$.
Furthermore we assume that for each $\sigma = (W\_\sigma, f\_\sigma)\in S$, the element $f\_\sigma \in k \langle X \rangle$ is a linear combination of monomials that are less than $W\_\sigma$.

The idea is that you replace (reduce) certain monomials (the $W\_\sigma$) with lower-order terms (the $f\_\sigma$).
A monomial $w \in \langle X \rangle$ is defined to be irreducible with respect to $S$ if $w$ does not contain any $W\_\sigma$ as a subword.
The classes of the irreducible monomials always span the algebra $A$; the Diamond Lemma gives a criterion (all ambiguities must be resolvable) in order for the classes of irreducible monomials to be independent.
In that case we say that $A$ is a PBW-algebra and that $X$ is a set of PBW-generators.

Specialization
--------------

Say that the algebra $A$ is specializable (at $0$) if each $f\_\sigma$ is of the form
$$
f\_\sigma = \sum\_{w \in \langle X \rangle} c\_\sigma^w w,
$$
where each coefficient $c\_\sigma^w$ lies in the local subring $k\_0 \subseteq k$ consisting of rational functions that do not have a pole at $0$.

Then we define a reduction system $S\_0$ for the specialized algebra $A\_0$ by simply evaluating all of the relations for $A$ at $t = 0$.
Formally, for each $\sigma = (W\_\sigma, f\_\sigma) \in S$ we define 
$$ \sigma\_0 = (W\_\sigma, f\_\sigma(0)) \in S\_0,$$
where
$$f\_\sigma(0) = \sum\_{w \in \langle X \rangle} c\_\sigma^w(0) w.$$
This is well-defined because we assumed that none of the rational functions $c\_\sigma^w$ had a pole at zero.
Finally, define the specialization of $A$ at $0$ to be the $\mathbb{C}$-algebra
$$
A\_0 = \mathbb{C} \langle X \rangle / (W\_\sigma - f\_\sigma(0)).
$$
Note that the specialized reduction system $S\_0$ has the same set of irreducible words as $S$ did.

Question
--------

If the irreducible words are independent in $A$, are they independent in $A\_0$?
Another way of putting it is this: if $X$ was a set of PBW-generators for the original $\mathbb{C}(t)$-algebra $A$, is it also a set of PBW-generators for the $\mathbb{C}$-algebra $A\_0$?