Let me say something about Type II identities.
These are the identities of the form $w\approx x$
where $x$ is a variable and $w=s(t_1,\ldots,t_n)$
where $s$ is a function symbol and each $t_i$ is a term
in a set $V$ of variables.
I will call the function symbol $s$ the outermost
function symbol on the LHS.
I take the question to be: Let $L$ be an algebraic language.
What can be said about an equational theory $\Sigma$
in the language $L$ if $\Sigma$ has a set
of axioms consisting of Type II identities?
I don't know the full answer, but I will make a few comments.
Observation.
An equational theory $\Sigma$ contains a
Type II identity $s(t_1,\ldots,t_n)\approx x$,
for a particular choice of the outermost function symbol $s$,
if and only if
$s$ interprets as a surjective function
in every model of $\Sigma$.
Only if. Assume that $\Sigma$ contains
$s(t_1,\ldots,t_n)\approx x$ and $M$ is a model of $\Sigma$.
We must show that $s^M\colon M^n\to M$ is a surjective function.
Choose $m\in M$ arbitrarily, let $v$ be any valuation
$v\colon V\to M$ such that $v(x)=m$.
Since $M\models s(t_1,\ldots,t_n)\approx x$, we have
$s^M(t_1[v],\ldots,t_n[v]) = x[v] = v(x) =m$, so $m$
is in the image of $s^M$. $\Box$
If.
Assume $s$ is surjective on all models of $\Sigma$.
Let $F = F_{\Sigma}(x)$ be the free $\Sigma$-algebra
over the set $\{x\}$.
Since $s^F\colon F^n\to F$ is surjective, there is
an $n$-tuple $(t_1,\ldots,t_n)\in F^n$ such that
$s^F(t_1,\ldots,t_n)=x\in F$. This relation
satisfied by the free generator $x\in F$
must hold universally in all models of $\Sigma$
by the universal mapping property for free algebras, so all
models of $\Sigma$ satisfy $s(t_1,\ldots,t_n)\approx x$. $\Box$
The above observation makes it clear that if
${\bf A}$ is an algebra that has no surjective
fundamental operations, then ${\bf A}$ does not satisfy
any Type II identities. This is enough to conclude that
some equational theories do not have axioms consisting
of Type II identities.
Groups have surjective multiplication
$\ast\colon G\times G\to G$ and surjective inverse
${}^{-1}\colon G\to G$,
raising the question of whether the theory of groups
can be axiomatized by Type II identities.
The answer is Yes. An old result about this is in
A. Tarski.
Ein Beitrag zur Axiomatik der Abelschen Gruppen.
Fundamenta Mathematicae, 30:253-256, 1938.
Tarski proves that the class of abelian groups
can be axiomatized in the language of division
$x/y: = x-y$ by the Type II identity
$$
x / (y / (z / (x / y))) \approx z.
$$
Higman and Neumann proved in
G. Higman and B. H. Neumann.
Groups as groupoids with one law.
Publicationes Mathematicae Debrecen, 2:215-227, 1952.
that the class of all (not necessarily abelian)
groups can also be axiomatized in the language of division
by the Type II identity
$$
x / ((((x / x) / y) / z) / (((x / x) / x) / z)) \approx y.
$$
Kunen proved in
K. Kunen.
Single axioms for groups.
J. Automated Reasoning, 9(3):291-308, 1992.
that the class of all groups can be axiomatized in the language
of multiplication and inverse by the Type II identity
$$
((z * (x * y)')' * (z * y')) * (y' * y)' \approx x.
$$
McCune proved in
W. McCune
Single axioms for Boolean algebra
Mathematics and Computer Science Division
Technical Memorandum No. 243, June 2000
that the class of Boolean algebras expressed in the language
of NOR (Sheffer stroke) is axiomatized by the Type II identity
$$(((((x|(((y |z )|(y |(u|z )))|y ))|y )|(((y |z )|(y |(u|z )))|(x|(y |
y ))))|((((v |(v |w ))|w )|(v |(v |(w |w ))))|y ))|y )|((((v |(v |w ))|
w )|(v |(v |(w |w ))))|((((x|(((y |z )|(y |(u|z )))|y ))|y )|(((y |
z )|(y |(u|z )))|(x|(y |y ))))|(y |y ))) \approx w.$$
All of the above results seem to focus on theories axiomatized
by a single Type II identity.
There is an interesting result that connects
theories axiomatized by finitely
many Type II identities with theories axiomatized by
a single Type II identity in
R. McKenzie
Equational bases for lattice theories.
Mathematica Scandinavica, Vol. 27, No. 1 (1970), pp. 24-38.
Theorem 1.2.
(paraphrased)
If $\Sigma$ is an equational theory that can be axiomatized
by finitely many Type II identities,
and $\Sigma$ contains the near unanimity identities for some term
$m(x_1,\ldots,x_k)$, then $\Sigma$ can be axiomatized
by a single Type II identity.
(The near unanimity identities for $m$ are
$m(y,x,x,\ldots,x)$ = $m(x,y,x,\ldots,x)$ = $\cdots$ = $m(x,x,x,\ldots,y)$ = $x$.)
