\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 each 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 a variety $\vr V$ satisfies no nontrivial
  idempotent Maltsev condition.  Is it true that $\vr L(\vr
  V) = \vr L$?  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 that 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 into 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 \hbox{$\beta\wedge\gamma =
  0$.}  Does the interval $I[\alpha,\theta]$ contain some
instance of abelianness or rectangulation? (Specifically,
must some 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 can be localized to single algebras, as Hobby and
McKenzie have done for finite algebras.

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{\sc Problem  P5.}
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.}
Is it true that every 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}. An affirmative solution to Problem P6
will remove the remaining obstacle to extending all of these
type omitting theorems to arbitrary varieties.

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{\sc Problem  P7.}
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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The weak term condition has been investigated in \cite{sym,
  k-kiss, k-kiss3}. Homomorphic images of finite Abelian
algebras are weakly Abelian and every residually small
variety generated by a finite nilpotent algebra is also
weakly Abelian. A better understanding of this concept could
be achieved by giving an equational-type characterization of
weakly Abelian varieties related to Theorem~7.7
of~\cite{k-kiss}.

