\chapter*{Problems}\label{problems}

In this aftersection, we collect all the problems from the text and 
list a few more. To learn the context of the first set of problems,
the reader should revisit that part of the text. 
In the second set we include a short
explanation of the background of the problem.

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\centerline{\bf From the text.}

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{\sc Problem \ref{1rst-order_ax}.}
Which congruence prevarieties are 
first-order axiomatizable?

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{\sc Problem \ref{1rst-order_ax_2}.}
Is the congruence prevariety of a congruence
modular variety first-order axiomatizable?

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{\sc Problem \ref{cong_quasi_maltsev}.}
Is the class of varieties satisfying a given
set of congruence quasi-identities
definable by (idempotent, linear) Maltsev conditions?

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{\sc Problem \ref{same_prevariety}.}
Suppose that $\vr V$ satisfies no nontrivial 
idempotent Maltsev condition.
Is it true that the congruence prevariety
$\vr L(\vr V)$ is the variety $\vr L$ of all lattices?
Is it true at least that $\vr L(\vr V)$ contains
the lattice $\Pi_4$ 
of all equivalence relations on a $4$-element set?

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{\sc Problem \ref{meet-continuous_prob}.}
Determine the number of meet continuous lattice varieties.

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{\sc Problem \ref{weakest_prb}.}
Is there a weakest congruence quasi-identity?

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{\sc Problem \ref{6.21}.}
Assume that $\vr V$ has a weak difference term,
and that $\m a\in {\vr V}$ is $\XXX$-solvable.
Does $\m a$ have a local Maltsev term operation?
Does $\m a$ have a Maltsev term operation?
If $\vr V$ is generated by $\XXX$--solvable algebras,
then must $\vr V$ be congruence permutable?

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{\sc Problem \ref{day-kiss_prb}.}
Show that if $\vr V$ is residually small 
and congruence modular, then $\vr V$
is congruence distributive or $\CON(\vr V)=\CON(\vr M)$
for some variety $\vr M$ of modules.

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{\sc Problem \ref{SI_mod}.}
Suppose that $\vr V$ satisfies a nontrivial
idempotent Maltsev condition, and every 
subdirectly irreducible member of $\vr V$
generates a congruence modular subvariety.
Must $\vr V$ be congruence modular?
(We are interested in the case where
$\vr V$ is residually large.)


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{\sc Problem \ref{9.12}.}
Suppose that $\vr V$ has a Taylor term.
If some algebra in $\vr V$ has a nontrivial
abelian tolerance,
must some subdirectly irreducible algebra in
$\vr V$ have a nontrivial abelian tolerance?

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\centerline{\bf Additional problems.}

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{\sc Problem   P1.}
Suppose that $\vr V$ satisfies no congruence identity.
Is it true the congruence prevariety of $\vr V$
contains the congruence prevariety of the variety
of semilattices? What if $\vr V$ has a join term?

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If $\vr V$ satisfies no nontrivial congruence identity,
then it contains some algebra $\m a$ that has
a nontrivial rectangular tolerance, $T$.
If $\vr V$ has a join term, then it acts 
as a compatible semilattice operation on $T$.
Can this fact be used to prove that the congruence lattice
of an arbitrary semilattice is embeddable the congruence
lattice of some subdirect
power of $\m a$? (This would give
an affirmative answer to the second part of Problem P1.)
If $\vr V$ does not have a join term, 
will a similar argument work using compatible relations in place
of the join term? 

If the answer to the first part of Problem P1
is affirmative, then the
congruence prevariety of the variety of semilattices
is a unique minimal congruence prevariety that generates
the variety of all lattices. This leads to the next 
question.

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{\sc Problem   P2.}
Determine all minimal congruence prevarieties.

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Could it be that every minimal congruence prevariety
is modular or equal to the congruence prevariety
of the variety of semilattices?

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{\sc Problem   P3.}
Are there only 
countably many join semidistributive congruence varieties?

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It is proved in Theorem 9 of \cite{hutchinson-czedli}
that there are continuum many different congruence varieties
contained in the congruence variety of abelian groups.
Problem P3 asks whether there are relatively few congruence 
varieties of varieties omitting abelian congruences.

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{\sc Problem  P4.}
Let $\m a$ be an arbitrary algebra. Suppose $\Conb(\m a)$
contains congruences such that 
$\alpha\vee\beta = \theta$,
$\alpha\vee\gamma = \theta$ and $\beta\wedge\gamma = 0$.
Does the interval $I[\alpha,\theta]$ contain some instance of 
abelianness or rectangulation? (Specifically, must some quotient
quotient $\m a/\delta$, $\alpha\leq \delta < \theta$ 
have a nontrivial abelian or rectangular tolerance?)
If so, how can it be identified?

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Lemma~5.19 of \cite{hobby-mckenzie} proves that
Problem P4 has an affirmative solution if $\m a$
is finite, namely join semidistributivity failures
involve nontrivial abelian or rectangular tolerances. 
Our Theorem~\ref{sdj-thm} proves that if
${\vr V}$ has no algebra with a nontrivial
abelian or rectangular tolerance, then it
has no algebra with a join semidistributivity failure.
Problem P4 asks whether our result be localized to single algebras,
as Hobby and McKenzie have done for finite algebras.

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{\sc Problem  P5.}
%Investigate nilpotence and solvability with respect to the rectangular
%commutator.
Investigate rectangular nilpotence and solvability in varieties
with a join term.
Can anything interesting be discovered about rectangularly
nilpotent or solvable varieties that have a join term?

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In this monograph we have examined only the `abelianness
concept' associated to rectangulation. Problem P5 suggests
the next step.

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%{\sc Problem  P6.}
%Develop and apply a commutator theory for the weak term condition.
%Is there a Klukovits-type characterization of weakly abelian 
%varieties?

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{\sc Problem   P6.}
Is it true that an idempotent variety that is not 
interpretable in the variety of distributive lattices
is congruence $n$-permutable for some $n$?

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Hobby and McKenzie have proved a number of type omitting
theorems for locally finite varieties
in Chapter~9 of \cite{hobby-mckenzie}. All two of them
of them have been strengthened and generalized to
arbitrary varieties. An affirmative solution to Problem P6 
will remove the remaining obstacle to extending all of the
type omitting theorems to arbitrary varieties.

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%{\sc Problem   P7.} (Kaarli, Pixley)
%Is there an almost congruence distributive variety?

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%We obtained partial results on this problem
%in Chapter 9.
