Theorem 4.23; equivalently it contradicts (1) and thus the proof is complete.
∎

######  Corollary 4.25.

Let $\mathsf{Q}$ be a quasivariety such that
${\mathbf{I}}({\textbf{F}}_{\mathsf{Q}})={\mathbf{I}}{\mathbf{P}}_{u}({\textbf{F}}_{\mathsf{Q}})$,
then the following are equivalent.

  1. (1)

$\mathsf{Q}$ is passively structurally complete;

  2. (2)

${\textbf{F}}_{\mathsf{Q}}\models\Sigma\Rightarrow\emptyset$ implies $\Sigma$
is trivializing in $\mathsf{Q}$;

  3. (3)

either ${\textbf{F}}_{\mathsf{Q}}$ is trivial, or $\Sigma$ is trivializing in
${\mathbf{Q}}({\textbf{F}}_{\mathsf{Q}})$ implies $\Sigma$ is trivializing in
$\mathsf{Q}$;

  4. (4)

every nontrivial finitely presented algebra is unifiable;

  5. (5)

every nontrivial algebra in $\mathsf{Q}$ is unifiable.

######  Remark 4.26.

The previous corollary can be applied whenever ${\textbf{F}}_{\mathsf{Q}}$ is
finite, therefore to all locally finite quasivarieties, but also to more
complex classes of algebras, e.g., all subquasivarieties of $\mathsf{FL}_{w}$
(see Subsection 5.2).

We will see an interesting application of Theorem 4.23 (or Corollary 4.25) in
substructural logics in Subsection 5.2; let us now show some other
consequences. Given a quasivariety $\mathsf{Q}$ whose smallest free algebra
${\textbf{F}}_{\mathsf{Q}}$ is nontrivial, let us consider the following set:

$\mathcal{P}_{\mathsf{Q}}=\\{\Sigma\Rightarrow\delta:\mathsf{Q}({\textbf{F}}_{\mathsf{Q}})\vDash\Sigma\Rightarrow\\{x\approx
y\\},\delta\mbox{ any identity}\\}.$

$\mathcal{P}_{\mathsf{Q}}$ axiomatizes a subquasivariety of $\mathsf{Q}$, that
we denote with $\mathsf{P}_{\mathsf{Q}}$. From Theorem 4.23 we get the
following.

######  Corollary 4.27.

Let $\mathsf{Q}$ be a quasivariety such that ${\textbf{F}}_{\mathsf{Q}}$ is
nontrivial. Every passive structurally complete subquasivariety of
$\mathsf{Q}$ is contained in $\mathsf{P}_{\mathsf{Q}}$, which is the largest
subquasivariety of $\mathsf{Q}$ that is passive structurally complete.

Moreover, for locally finite quasivarieties the characterization theorem reads
as follows.

######  Corollary 4.28.

Let $\mathsf{Q}$ be a locally finite quasivariety, then the following are
equivalent.

  1. (1)

$\mathsf{Q}$ is passive structurally complete;

  2. (2)

every algebra in $\mathsf{Q}$ is unifiable;

  3. (3)

every finite algebra in $\mathsf{Q}$ is unifiable;

  4. (4)

every finite subdirectly irreducible in $\mathsf{Q}$ is unifiable.

A nontrivial algebra A is _Kollár_ if it has no trivial subalgebras, and a
quasivariety $\mathsf{Q}$ is a Kollár quasivariety if all nontrivial algebras
in $\mathsf{Q}$ are Kollár. By [58] if A belongs to a Kollár quasivariety,
$1_{\textbf{A}}$, the largest congruence of A, is compact in
$\textrm{Con}_{\mathsf{Q}}({\textbf{A}})$; from there a straightforward
application of Zorn’s Lemma yields that if A is nontrivial there is at least
one maximal congruence $\theta\in\textrm{Con}_{\mathsf{Q}}({\textbf{A}})$
below $1_{\textbf{A}}$ (i.e. ${\textbf{A}}/\theta$ is relative simple).

######  Theorem 4.29.

If $\mathsf{Q}$ is a Kollár quasivariety and ${\textbf{F}}_{\mathsf{Q}}$ is
the only finitely generated relative simple algebra in $\mathsf{Q}$, then
$\mathsf{Q}$ is passive structurally complete.

###### Proof.

Let A be a nontrivial finitely presented algebra in $\mathsf{Q}$; since
$\mathsf{Q}$ is a Kollár quasivariety, A has a relative simple homomorphic
image, that must be finitely generated. Hence it must be equal to
${\textbf{F}}_{\mathsf{Q}}$, so A is unifiable; by Theorem 4.23 $\mathsf{Q}$
is passive structurally complete. ∎

######  Corollary 4.30.

For a locally finite Kollár quasivariety $\mathsf{Q}$ such that
${\textbf{F}}_{\mathsf{Q}}$ has no proper subalgebra the following are
equivalent:

  1. (1)

${\textbf{F}}_{\mathsf{Q}}$ is the only finite relative simple algebra in
$\mathsf{Q}$;

  2. (2)

$\mathsf{Q}$ is passive structurally complete.

###### Proof.

If (1) holds, than (2) holds by Theorem 4.29. Conversely assume (2); then
every nontrivial finitely presented algebra in $\mathsf{Q}$ is unifiable.
Since $\mathsf{Q}$ is locally finite ${\textbf{F}}_{\mathsf{Q}}$ is finite and
nontrivial since $\mathsf{Q}$ is Kollár; now since ${\textbf{F}}_{\mathsf{Q}}$
has no proper subalgebra no finite relative simple algebra different from
${\textbf{F}}_{\mathsf{Q}}$ can be unifiable, but $\mathsf{Q}$ must contain at
least a relative simple algebra [50, Theorem 3.1.8]. Hence
${\textbf{F}}_{\mathsf{Q}}$ must be relative simple and (1) holds. ∎

The next results will allow us to find interesting applications in varieties
of bounded lattices, which we will explore in Section 5.1. We say that an
algebra A in a variety $\mathsf{V}$ is flat if
${\mathbf{H}}{\mathbf{S}}({\textbf{A}})$ does not contain any simple algebra
different from ${\textbf{F}}_{\mathsf{V}}$.

######  Theorem 4.31.

Let $\mathsf{V}$ be a Kollár variety; if every finitely generated algebra in
$\mathsf{V}$ is flat then $\mathsf{V}$ is passive structurally complete. If
$\mathsf{V}$ is locally finite, then the converse holds as well.

###### Proof.

First, if ${\textbf{F}}_{\mathsf{V}}$ is trivial then $\mathsf{V}$ is
vacuously passive structurally complete. If ${\textbf{F}}_{\mathsf{V}}$ is
nontrivial and every finitely generated algebra is flat, then the only
finitely generated simple lattice in $\mathsf{V}$ must be
${\textbf{F}}_{\mathsf{V}}$; since $\mathsf{V}$ is Kollár, $\mathsf{V}$ is
passive structurally complete by Theorem 4.29.

If $\mathsf{V}$ is locally finite and passive structurally complete, then
${\textbf{F}}_{\mathsf{V}}$ is the only finite simple algebra in $\mathsf{V}$
by Corollary 4.30. It follows that no finite simple algebra different from
${\textbf{F}}_{\mathsf{V}}$ can appear in
${\mathbf{H}}{\mathbf{S}}({\textbf{A}})$ for any finite
${\textbf{A}}\in\mathsf{V}$. So every finite algebra in $\mathsf{V}$ must be
flat. ∎

######  Theorem 4.32.

Let $\mathsf{V}$ be a congruence distributive Kollár variety; a finitely
generated variety $\mathsf{W}\subseteq\mathsf{V}$ is passive structurally
complete if and only if each generating algebra is flat.

###### Proof.

Suppose that $\mathsf{W}={\mathbf{V}}(K)$ where $K$ is a finite set of finite
algebras; by Jónsson Lemma any simple algebra in $\mathsf{V}$ is in
${\mathbf{H}}{\mathbf{S}}(K)$. If $K$ consists entirely of flat algebras, then
there cannot be any simple algebra in $\mathsf{V}$ different from
${\textbf{F}}_{\mathsf{W}}$, so $\mathsf{W}$ is passive structurally complete.
On the other hand if ${\textbf{A}}\in K$ is not flat, then there is an algebra
${\textbf{B}}\in{\mathbf{H}}{\mathbf{S}}(K)$ which is simple and different
from ${\textbf{F}}_{\mathsf{V}}$. Clearly ${\textbf{B}}\in\mathsf{W}$, which
is not passive structurally complete. ∎

##  5\. Applications to algebra and logic

In this last section we will see some relevant examples and applications of
our results in the realm of algebra and (algebraic) logic that deserve a
deeper exploration than the examples already presented in the previous
sections. We will start with focusing on varieties of lattices and bounded
lattices, and then move to their expansions that are the equivalent algebraic
semantics of subtructural logics: residuated lattices.

As a main result, in the last subsection we present the logical counterpart of
the characterization of passive structural completeness in substructural
logics with weakening, that is, such a logic is passively structurally
complete if and only if every classical contradiction is explosive in it;
building on this, from the algebraic perspective, we are able to axiomatize
the largest variety of representable bounded commutative integral residuated
lattices that is passively structurally complete (and such that all of its
quasivarieties have this property). Notice that this characterization
establishes negative results as well: if a logic (or a quasivariety) is not
passively structurally complete, a fortiori it is not structurally complete
either.

###  5.1. (Bounded) lattices

In this subsection we start with some results about primitive (quasi)varieties
of lattices, and then move to bounded lattices, where in particular we obtain
some new results about passive structurally complete varieties.

####  5.1.1. Primitivity in lattices

Many examples of quasivarieties that are primitive can be found in lattices
satisfying Whitman’s condition (W); Whitman’s condition is a universal
sentence that holds in free lattices:

(W) $\\{x\wedge y\leq u\vee v\\}\Rightarrow\\{x\leq u\vee v,y\leq u\vee
v,x\wedge y\leq u,x\wedge y\leq v\\}.$

Now a finite lattice is finitely projective in the variety of all lattices if
and only if it satisfies (W) [33], which implies:

######  Lemma 5.1.

Let $\mathsf{K}$ be a finite set of finite lattices. If every lattice in
$\mathsf{K}$ satisfies (W) then ${\mathbf{Q}}(\mathsf{K})$ is primitive.

###### Proof.

${\mathbf{Q}}(\mathsf{K})$ is locally finite and by Theorem 2.3(2) every
relative subdirectly irreducible lies in
${\mathbf{I}}{\mathbf{S}}(\mathsf{K})$; as (W) is a universal sentence it is
preserved under subalgebras, thus they all satisfy (W) and hence they are all
finitely projective in the variety of lattices and then also in
${\mathbf{Q}}(\mathsf{K})$. By Theorem 4.11(4), ${\mathbf{Q}}(\mathsf{K})$ is
primitive. ∎

Luckily finite lattices satisfying (W) abound, so there is no shortage of
primitive quasivarieties of lattices. For varieties of lattices the situation
is slightly different; in particular, because of Lemma 4.4 it is not enough
that all lattices in $\mathsf{K}$ are weakly projective in
${\mathbf{V}}(\mathsf{K})$ to guarantee that ${\mathbf{V}}(\mathsf{K})$ is
structurally complete.

First we introduce some lattices: ${\textbf{M}}_{n}$ for $3\leq n\leq\omega$
are the modular lattices consisting of a top, a bottom, and $n$ atoms while
the lattices ${\textbf{M}}_{3,3}$ and ${\textbf{M}}_{3,3}^{+}$ are displayed
in Figure 4.

${\textbf{M}}_{3,3}$${\textbf{M}}_{3,3}^{+}$ Figure 4. ${\textbf{M}}_{3,3}$
and ${\textbf{M}}_{3,3}^{+}$

Observe that all the above lattices, with the exception of
${\textbf{M}}_{3,3}$, satisfy (W). Now Gorbunov ([50], Theorem 5.1.29) showed
that ${\textbf{M}}^{+}_{3,3}$ is splitting in the lattice of subquasivarieties
of modular lattices. More in detail for any quasivariety $\mathsf{Q}$ of
modular lattices, either ${\textbf{M}}^{+}_{3,3}\in\mathsf{Q}$ or else
$\mathsf{Q}={\mathbf{Q}}({\textbf{M}}_{n})$ for some $n\leq\omega$. Observe
that, for $n<\omega$, $\mathsf{Q}({\textbf{M}}_{n})$ is primitive by Lemma 5.1
and ${\mathbf{V}}({\textbf{M}}_{n})={\mathbf{Q}}({\textbf{M}}_{n})$ by Lemma
2.4; then the only thing left to show is that
${\mathbf{V}}({\textbf{M}}_{\omega})$ is a primitive variety and Gorbunov did
exactly that. On the other hand no variety $\mathsf{V}$ of lattices containing
${\textbf{M}}_{3,3}^{+}$ can be primitive; in fact ${\textbf{M}}_{3,3}$ is a
simple homomorphic image of ${\textbf{M}}^{+}_{3,3}$ that cannot be embedded
in ${\textbf{M}}^{+}_{3,3}$. By Lemma 2.4,
${\mathbf{Q}}({\textbf{M}}_{3,3}^{+})\subsetneq{\mathbf{V}}({\textbf{M}}_{3,3}^{+})$,
so $\mathsf{V}$ contains a strict (i.e. not a variety) subquasivariety and
cannot be primitive. Thus Gorbunov’s result can be formulated as: a variety of
modular lattices is primitive if and only if it does not contain
${\textbf{M}}^{+}_{3,3}$. Note that it cannot be improved to quasivarieties:
since ${\textbf{M}}_{3,3}^{+}$ satisfies (W),
${\mathbf{Q}}({\textbf{M}}^{+}_{3,3})$ is primitive by Lemma 5.1. However we
observe:

######  Lemma 5.2.

If $\mathsf{Q}$ is a quasivariety of modular lattices and
${\textbf{M}}_{3,3}\in\mathsf{Q}$, then $\mathsf{Q}$ is not primitive.

###### Proof.

Clearly the two element lattice $\mathbf{2}\in\mathsf{Q}$ and it is easy to
check that ${\textbf{M}}_{3,3}^{+}\leq_{sd}\mathbf{2}\times{\textbf{M}}_{3,3}$
so ${\textbf{M}}_{3,3}^{+}\in\mathsf{Q}$ and
${\textbf{M}}_{3,3}\in{\mathbf{H}}({\textbf{M}}_{3,3}^{+})$. Since
${\textbf{M}}_{3,3}$ cannot be embedded in ${\textbf{M}}_{3,3}^{+}$, in
$\mathsf{Q}$ there is a simple finite (so finitely presented, since lattices
have finite type) algebra that is not weakly projective. By Theorem 4.11,
$\mathsf{Q}$ is not primitive. ∎

Therefore to find a variety of modular lattices that is structurally complete
but not primitive it is enough to find a finite lattice F such that
${\textbf{M}}_{3,3}^{+}\in{\mathbf{V}}({\textbf{F}})$ but
$\mathsf{K}=\\{{\textbf{F}}\\}$ satisfies the hypotheses of Lemma 4.4. Bergman
in [13] observed that the Fano lattice F has exactly those characteristics;
the Fano lattice is the (modular) lattice of subspaces of
$(\mathbb{Z}_{2})^{3}$ seen as a vector space on $\mathbb{Z}_{2}$ and it is
displayed in Figure 5.

Figure 5. The Fano lattice

Now:

  1. $\bullet$

F is projective in ${\mathbf{V}}({\textbf{F}})$ [52];

  2. $\bullet$

the subdirectly irreducible members of ${\mathbf{V}}({\textbf{F}})$ are
exactly $\mathbf{2},{\textbf{M}}_{3},{\textbf{M}}_{3,3},{\textbf{F}}$ and they
are all subalgebras of F.

It follows that F does not satisfies (W) (since ${\textbf{M}}_{3,3}$ does
not), ${\mathbf{V}}({\textbf{F}})$ is structurally complete and (since
${\textbf{M}}_{3,3}\in{\mathbf{V}}({\textbf{F}})$) not primitive by Lemma 5.2;
also ${\mathbf{Q}}({\textbf{F}})$ is structurally complete but, since
${\textbf{M}}_{3,3}\in{\mathbf{Q}}({\textbf{F}})$, it cannot be primitive as
well.

Primitive varieties of lattices have been studied in depth in [53]; there the
authors proved the following theorem that explains the behavior we have seen
above.

######  Theorem 5.3 ([53]).

If A is a lattice satisfying (W), then ${\mathbf{V}}({\textbf{A}})$ is
primitive if and only if every subdirectly irreducible lattice in
${\mathbf{H}}{\mathbf{S}}({\textbf{A}})$ satisfies (W).

We believe that many of the techniques in [53] could be adapted to gain more
understanding of primitive quasivarieties of lattices, but proceeding along
this path would make this part too close to being a paper in lattice theory,
and we have chosen a different focus. We only borrow an example from [53] that
shows that Lemma 5.1 cannot be inverted for quasivarieties. Let
${\textbf{H}}^{+},{\textbf{H}}$ be the lattices in Figure 6.

It is easily seen that the pair ${\textbf{H}}^{+},{\textbf{H}}$ behaves almost
like the pair ${\textbf{M}}_{3,3}^{+},{\textbf{M}}_{3,3}$: ${\textbf{H}}^{+}$
satisfies (W) (so ${\mathbf{Q}}({\textbf{H}}^{+})$ is primitive), H does not
satisfy (W) and ${\textbf{H}}^{+}\leq_{sd}\mathbf{2}\times{\textbf{H}}$. As
above we can conclude that ${\mathbf{V}}({\textbf{H}}^{+})$ is not primitive.
However ${\mathbf{V}}({\textbf{H}})$ is primitive [53] so
${\mathbf{Q}}({\textbf{H}})$ is a primitive quasivariety generated by a finite
lattice not satisfying (W).

H${\textbf{H}}^{+}$ Figure 6. H and ${\textbf{H}}^{+}$

####  5.1.2. Bounded lattices

We now focus on applications of our results in varieties of bounded lattices.
A bounded lattice is a lattice with two constants, $0$ and $1$, that represent
the top and the bottom of the lattice. Bounded lattices form a variety
$\mathsf{L}^{b}$ that shares many features with variety of lattices. In
particular, let $\mathbf{2}^{b}$ be the two element bounded lattice, then the
variety of bounded distributive lattices is
$\mathsf{D}^{b}={\mathbf{I}}{\mathbf{S}}{\mathbf{P}}(\mathbf{2}^{b})$.
Therefore

${\mathbf{Q}}({\textbf{F}}_{\mathsf{D}_{b}}(\omega))\subseteq\mathsf{D}^{b}={\mathbf{I}}{\mathbf{S}}{\mathbf{P}}(\mathbf{2}^{b})\subseteq{\mathbf{Q}}({\textbf{F}}_{\mathsf{D}_{b}}(\omega))$

and by Theorem 4.1, the variety of bounded distributive lattices
$\mathsf{D}^{b}$ is structurally complete, as shown in [35]. In [14] it is
shown that locally finite, congruence modular, minimal varieties are
q-minimal; since these hypotheses apply to $\mathsf{D}^{b}$, the latter is
also primitive. However, it is not non-negative universally complete; it is a
nice exercise in general algebra to show that for any variety $\mathsf{V}$ of
bounded lattices, $1$ is join irreducible in
${\textbf{F}}_{\mathsf{V}}(\omega)$. It follows that

$\\{x\vee y\approx 1\\}\Rightarrow\\{x\approx 1,y\approx 1\\}$

is an active universal sentence that is admissible in $\mathsf{V}$. But it is
clearly not derivable, since any nontrivial variety of bounded lattices
contains $\mathbf{2}^{b}\times\mathbf{2}^{b}$ which does not satisfy the
universal sentence.

######  Proposition 5.4.

No nontrivial variety of bounded lattices is active universally complete.

Actually something more is true; if $\mathsf{V}$ is a variety of bounded
lattices that is structurally complete, then by Theorem 4.6, each finite
subdirectly irreducible algebra ${\textbf{A}}\in\mathsf{V}$ must satisfy the
above universal sentence, i.e. $1$ must be join irreducible in A. But the
bounded lattices ${\textbf{N}}_{5}^{b}$ and ${\textbf{M}}_{3}^{b}$ do not
satisfy that, so any structurally complete variety of bounded lattice must
omit them both. As in the unbounded case, this means that the variety must be
the variety of bounded distributive lattices. Thus:

######  Proposition 5.5 ([35]).

The variety of bounded distributive lattices is the only (active) structurally
complete variety of bounded lattices.

We have seen that active structural completeness does not have much meaning in
bounded lattices. Passive structural completeness has more content, as we are
now going to show. Notice that any variety of bounded lattices is Kollár and
${\textbf{F}}_{\mathsf{V}}=\mathbf{2}^{b}$ for any variety $\mathsf{V}$ of
bounded lattices. Since $\mathbf{2}^{b}$ is simple and has no proper
subalgebras, any simple bounded lattice not isomorphic with $\mathbf{2}^{b}$
is not unifiable; in particular if a variety $\mathsf{V}$ contains a finite
simple lattice L different from $\mathbf{2}^{b}$, then
${\mathbf{V}}({\textbf{L}})$ cannot be passive structurally completeby
Corollary 4.30, and hence neither can $\mathsf{V}$.

We will use this fact to show that the only variety of bounded modular
lattices that is passive structurally complete is the one we already know to
possess that property, i.e. the variety $\mathsf{D}^{b}$ of bounded
distributive lattices. A key step is to show that ${\textbf{M}}^{b}_{3}$ is
splitting in the variety of bounded modular lattices; in the unbounded case,
this follows from the fact that ${\textbf{M}}_{3}$ is projective and
subdirectly irreducible. However, ${\textbf{M}}_{3}^{b}$ is not projective in
the variety of bounded modular lattices. Indeed, the lattice in Figure 5.1.2
is a bounded modular lattice having ${\textbf{M}}^{b}_{3}$ as homomorphic
image, but it has no subalgebra isomorphic with ${\textbf{M}}^{b}_{3}$, which
hence cannot be a retract.

$0$$1$

However we can use A. Day idea in [34]; a finite algebra A is finitely
projected in a variety $\mathsf{V}$ if for any ${\textbf{B}}\in\mathsf{V}$ if
$f:{\textbf{B}}\longrightarrow{\textbf{A}}$ is surjective, then there is a
finite subalgebra C of B with $f({\textbf{C}})\cong{\textbf{A}}$. Clearly any
finite projective lattice is finitely projected. A finite algebra A splitting
in a variety $\mathsf{V}$ if ${\textbf{A}}\in\mathsf{V}$ and there is a
subvariety $\mathsf{W}_{\textbf{A}}\subseteq\mathsf{V}$ such that for any
variety $\mathsf{U}\subseteq\mathsf{V}$ either ${\textbf{A}}\in\mathsf{U}$ or
$\mathsf{U}\subseteq\mathsf{W}_{\textbf{A}}$. This simply means that the
lattice of subvarieties of $\mathsf{V}$ is the disjoint union of the filter
generated by ${\mathbf{V}}({\textbf{A}})$ and the ideal generated by
$\mathsf{W}_{\textbf{A}}$. The key result is:

######  Theorem 5.6.

([34], Theorem 3.7) If $\mathsf{V}$ is a congruence distributive variety, then
any finitely projected subdirectly irreducible algebra in $\mathsf{V}$ is
splitting in $\mathsf{V}$.

######  Lemma 5.7.

Let $\mathsf{V}^{b}$ be a variety of bounded lattices and let $\mathsf{V}$ be
the variety of lattice subreducts of $\mathsf{V}^{b}$. If L is finitely
projected in $\mathsf{V}$, then ${\textbf{L}}^{b}$ is finitely projected in
$\mathsf{V}^{b}$.

###### Proof.

The fact that $\mathsf{V}$ is indeed a variety is easy to check. Let now
${\textbf{A}}^{b}\in\mathsf{V}^{b}$ and suppose that there is an onto
homomorphism $f:{\textbf{A}}^{b}\longrightarrow{\textbf{L}}^{b}$; then $f$ is
onto from A to L and since L is finitely projected in $\mathsf{V}$ there is a
subalgebra B of A with $f({\textbf{B}})\cong{\textbf{L}}$. But
$B\cup\\{0,1\\}$ is the universe of a finite subalgebra C of
${\textbf{A}}^{b}$. Extend $f$ to $\hat{f}$ by setting $\hat{f}(0)=0$ and
$\hat{f}(1)=1$; then $\hat{f}({\textbf{C}})\cong{\textbf{L}}^{b}$ and so
${\textbf{L}}^{b}$ is finitely projected in $\mathsf{V}^{b}$. ∎

######  Theorem 5.8.

A variety of modular bounded lattices is passive structurally complete if and
only if it is the variety of bounded distributive lattices.

###### Proof.

$\mathsf{D}^{b}$ is structurally complete, hence passive structurally
complete. Conversely observe that ${\textbf{M}}_{3}$ is projective in the
variety of modular lattices, so ${\textbf{M}}_{3}^{b}$ is finitely projected
in the variety of bounded modular lattices. Hence, by Theorem 5.6,
${\textbf{M}}^{b}_{3}$ is splitting in the variety, which means that for any
variety $\mathsf{V}$ of bounded modular lattices, either
${\textbf{M}}_{3}^{b}\in\mathsf{V}$ or $\mathsf{V}$ is $\mathsf{D}^{b}$. But
if ${\textbf{M}}^{b}_{3}\in\mathsf{V}$ then $\mathsf{V}$ cannot be passive
universally complete, since ${\textbf{M}}_{3}^{b}$ is simple. The conclusion
follows. ∎

In order to find other relevant varieties of bounded lattices that are passive
structurally complete, we are going to take a closer look at flat lattices.
Finding flat bounded lattices is not hard since the lattice of subvarieties of
lattices has been studied thoroughly and a lot is known about it (an excellent
survey is [54]). Clearly ${\textbf{N}}_{5}$ is flat and hence so is
${\textbf{N}}_{5}^{b}$; however we know exactly all the covers of the minimal
nondistributive varieties of lattices (which is of course
${\mathbf{V}}({\textbf{N}}_{5})$). There are 15 finite subdirectly irreducible
nonsimple lattices, commonly called ${\textbf{L}}_{1},\dots,{\textbf{L}}_{15}$
(some of them are in Figure 7) that generate all the join irreducible (in the
lattice of subvarieties) covers of ${\mathbf{V}}({\textbf{N}}_{5})$. It is
easy to see their bounded versions all are join irreducible covers of
${\mathbf{V}}({\textbf{N}}^{b}_{5})$ in the lattice of subvarieties of bounded
lattices. We suspect that they are also the only join irreducible covers; one
needs only to check that the (rather long) proof for lattices [57] goes
through for bounded lattices but we leave this simple but tedious task to the
reader. In any case for $i=1,\dots,15$ the subdirectly irreducible algebras in
${\mathbf{V}}({\textbf{L}}_{i}^{b})$ are exactly
$\mathbf{2}^{b},{\textbf{N}}_{5}^{b}$ and ${\textbf{L}}_{i}^{b}$ (via a
straightforward application of Jónsson Lemma); so each ${\textbf{L}}_{i}^{b}$
is flat and each ${\mathbf{V}}({\textbf{L}}_{i}^{b})$ is passively
structurally complete (by Theorem 4.32).

${\textbf{L}}_{13}$${\textbf{L}}_{14}$${\textbf{L}}_{15}$${\textbf{L}}_{1}$${\textbf{L}}_{2}$
Figure 7.

Let’s make more progress: consider the rules

($SD_{\wedge}$) $\displaystyle x\wedge y\approx x\wedge z\quad\Rightarrow\quad
x\wedge y\approx x\wedge(y\vee z)$ ($SD_{\vee}$) $\displaystyle x\vee y\approx
x\vee z\quad\Rightarrow\quad x\vee y\approx x\vee(y\wedge z).$

A lattice is meet semidistributive if it satisfies $SD_{\wedge}$, join
semidistributive if it satisfies $SD_{\wedge}$ and semidistributive if it
satisfies both. Clearly (meet/join) semidistributive lattices form
quasivarieties called $\mathsf{SD}_{\wedge}$, $\mathsf{SD}_{\vee}$ and
$\mathsf{SD}$ respectively, and so do their bounded versions. It is a standard
exercise to show that homomorphic images of a finite (meet/join)
semidistributive lattices are (meet/join) semidistributive. It is also
possible to show none of the three quasivariety (and their bounded versions)
is a variety (see [54] p. 82 for an easy argument); they are also not locally
finite since for instance ${\textbf{F}}={\textbf{F}}_{\mathsf{SD}}(x,y,z)$ is
infinite; hence ${\textbf{F}}^{b}$ is a bounded infinite three-generated
lattice and thus $\mathsf{SD}^{b}$ is not locally finite as well. A variety
$\mathsf{V}$ of (bounded) lattices is (meet/join) semidistributive if
$\mathsf{V}\subseteq\mathsf{SD}$ ($\mathsf{V}\subseteq\mathsf{SD}/\wedge$ /
$\mathsf{V}\subseteq\mathsf{SD}/\vee$).

We need a little bit of lattice theory. A filter of L is an upset $F$ of L
that is closed under meet; a filter is prime if $a\vee b\in F$ implies $a\in
F$ or $b\in F$. An ideal $I$ of L is the dual concept, i.e. a downset that is
closed under join; an ideal is prime if $a\wedge b\in I$ implies $a\in I$ or
$b\in I$. The following lemma is straightforward.

######  Lemma 5.9.

If $F$ is a prime filter of L ($I$ is a prime ideal of L), then $L\setminus F$
is a prime ideal of L (${\textbf{L}}\setminus I$ is a prime filter of L).

######  Lemma 5.10.

Any bounded (meet/join) semidistributive lattice is unifiable in the variety
of bounded lattices.

###### Proof.

Let L be bounded and meet semidistributive. Since L is lower bounded by $0$ a
standard application of Zorn Lemma yields a maximal proper filter $F$ of L; we
claim that $F$ is also prime. Let $a,b\notin F$; then the filter generated by
$F\cup\\{a\\}$ must be the entire lattice. Hence there must be a $c\in F$ with
$c\wedge a=0$; similarly there must be a $d\in F$ with $d\wedge b=0$. Let
$e=c\wedge d$; then $e\in F$ and $e\wedge a=e\wedge b=0$ and by meet
semidistributivity $e\wedge(a\vee b)=0$. But if $a\vee b\in F$, then $0\in F$,
a clear contradiction. Hence $a\vee b\notin F$ and $F$ is prime.

Let now $\varphi:{\textbf{L}}\Longrightarrow\mathbf{2}^{b}$ defined by

$\varphi(x)=\left\\{\begin{array}[]{ll}1,&\hbox{if $x\in F$;}\\\ 0,&\hbox{if
$x\notin F$.}\end{array}\right.$

Using the fact that $F$ is prime and $L\setminus F$ is prime it is
straightforward to check that $\varphi$ is a homomorphism. Therefore L is
unifiable.

A dual proof shows that the conclusion holds for join semidistributivity and a
fortiori for semidistributivity. ∎

######  Proposition 5.11.

Any bounded finite (meet/join) semidistributive lattice is flat.

###### Proof.

If L is finite and (meet/join) semidistributive, every lattice in
${\mathbf{H}}{\mathbf{S}}({\textbf{L}})$ is finite and (meet/join)
semidistributive. So it is unifiable and, if simple, it must be equal to
${\textbf{2}}^{b}$; therefore L is flat. ∎

######  Corollary 5.12.

Every locally finite (meet/join) semidistributive variety of bounded lattices
is passive structurally complete.

In [61] several (complex) sets of equations implying semidistributivity are
studied; one of them is useful to us, since it describes a class of locally
finite varieties. The description is interesting in that involves some of the
${\textbf{L}}_{i}^{\prime}s$ we have introduced before.

######  Theorem 5.13.

[61] There exists a finite set $\Gamma$ of lattices equations such that, if
$\mathsf{V}$ is any variety of lattices such that $\mathsf{V}\vDash\Gamma$,
then the following hold:

  1. (1)

$\mathsf{V}$ is semidistributive;

  2. (2)

$\mathsf{V}$ is locally finite;

  3. (3)

only ${\textbf{L}}_{13},{\textbf{L}}_{14},{\textbf{L}}_{15}\in\mathsf{V}$.

A variety satisfying $\Gamma$ is called almost distributive and it is
straightforward to check that a similar result holds for varieties of bounded
lattices. Therefore:

######  Proposition 5.14.

Every almost distributive variety of bounded lattices is passive structurally
complete.

We close this subsection with a couple of observations; first
${\mathbf{V}}({\textbf{L}}_{1}^{b},{\textbf{L}}_{2}^{b})$ is a variety of
bounded lattices that is passive structurally complete (by Theorem 4.32) but
neither meet nor join semidistributive. Next, what about infinite flat
(bounded) lattices? We stress that in [63] there are several examples of this
kind and we believe that a careful analysis of the proofs therein could give
some insight on how to construct a non locally finite variety of bounded
lattices that it is passive structurally complete. But again, this is not a
paper in lattice theory; therefore we defer this investigation.

###  5.2. Substructural logics and residuated lattices

Originally, substructural logics were introduced as logics which, when
formulated as Gentzen-style systems, lack some (including “none” as a special
case) of the three basic structural rules (i.e. exchange, weakening and
contraction) of classical logic. Nowadays, substructural logics are usually
intended as those logics whose equivalent algebraic semantics are residuated
structures, and they encompass most of the interesting non-classical logics:
intuitionistic logic, basic logic, fuzzy logics, relevance logics and many
other systems. Precisely, by substructural logics we mean here the axiomatic
extensions of the Full Lambek calculus $\mathcal{FL}$ (see [43] for details
and a survey on substructural logics). All these logics are strongly
algebraizable: their equivalent algebraic semantics are all varieties of
$\mathsf{FL}$-algebras, particular residuated lattices that we shall now
define.

A residuated lattice is an algebra ${\textbf{A}}=\langle
A,\vee,\wedge,\cdot,{/},{\backslash},1\rangle$ where

  1. (1)

$\langle A,\vee,\wedge\rangle$ is a lattice;

  2. (2)

$\langle A,\cdot,1\rangle$ is a monoid;

  3. (3)

${/}$ and ${\backslash}$ are the right and left divisions w.r.t. $\cdot$,
i.e., $x\cdot y\leq z$ iff $y\leq x{\backslash}z$ iff $x\leq z{/}y$, where
$\leq$ is given by the lattice ordering.

Residuated lattices form a variety $\mathsf{RL}$ and an equational
axiomatization, together with many equations holding in these very rich
structures, can be found in [19].

A residuated lattice A is integral if it satisfies the equation $x\leq 1$; it
is commutative if $\cdot$ is commutative, and in this case the divisions
coincide: $x\backslash y=y/x$, and they are usually denoted with $x\to y$. The
classes of residuated lattices that satisfy any combination of integrality and
commutativity are subvarieties of $\mathsf{RL}$. We shall call the variety of
integral residuated lattices $\mathsf{IRL}$, commutative residuated lattices
$\mathsf{CRL}$, and their intersection $\mathsf{CIRL}$.

Residuated lattices with an extra constant $0$ in the language are called
$\mathsf{FL}$-algebras, since they are the equivalent algebraic semantics of
the Full Lambek calculus $\mathcal{FL}$. Residuated lattices are then the
equivalent algebraic semantics of $0$-free fragment of $\mathcal{FL}$,
$\mathcal{FL}^{+}$. An $\mathsf{FL}$-algebra is _$0$ -bounded_ if it satisfies
the inequality $0\leq x$ and the variety of zero-bounded
$\mathsf{FL}$-algebras is denoted by $\mathsf{FL}_{o}$; integral and
$0$-bounded $\mathcal{FL}$-algebras are called $\mathcal{FL}_{w}$ algebras
(since they are the equivalent algebraic semantics of the Full Lambek Calculus
with weakening), and we call its commutative subvariety $\mathcal{FL}_{ew}$.

Restricting ourselves to the commutative case there is another interesting
equation:

$(x\rightarrow y)\vee(y\rightarrow x)\approx 1.$

It can be shown (see [19] and [55]) that a subvariety of $\mathsf{FL_{ew}}$ or
$\mathsf{CIRL}$ satisfies the above equation if and only if any algebra
therein is a subdirect product of totally ordered algebras, and this implies
that all the subdirectly irreducible algebras are totally ordered. Such
varieties are called representable and the subvariety axiomatized by that
equation is the largest subvariety of $\mathsf{FL_{ew}}$ or $\mathsf{CIRL}$
that is representable. The representable subvariety of $\mathsf{FL}_{ew}$ is
usually denoted by $\mathsf{M}\mathsf{T}\mathsf{L}$, since it is the
equivalent algebraic semantics of Esteva-Godo’s Monoidal t-norm based logic
[38].

####  5.2.1. Active universal completeness

We have already seen examples of subvarieties of $\mathsf{FL}_{ew}$-algebras
that are active universally complete, but those were all locally finite
subvarieties of $\mathsf{BL}$-algebras, that is, $\mathsf{MTL}$-algebras
satisfying the divisibility equation: $x\land y=x(x\to y)$. In this section we
will display a different class of examples. If A is any algebra a congruence
$\theta\in\operatorname{Con}({\textbf{A}})$ is a factor congruence if there is
a $\theta^{\prime}\in\operatorname{Con}({\textbf{A}})$ such that
$\theta\vee\theta^{\prime}=1_{\textbf{A}}$,
$\theta\wedge\theta^{\prime}=0_{\textbf{A}}$ and $\theta,\theta^{\prime}$
permute. It is an easy exercise in general algebra to show that in this case
${\textbf{A}}\cong{\textbf{A}}/\theta\times{\textbf{A}}/\theta^{\prime}$; note
that $1_{\textbf{A}}$ and $0_{\textbf{A}}$ are factor congruences that gives a
trivial decomposition. A less known fact (that appears in [31]) is:

######  Lemma 5.15.

Let A be any algebra and $\theta$ a factor congruence; then
${\textbf{A}}/\theta$ is a retract of A if and only if there is a homomorphism
$h:{\textbf{A}}/\theta\longrightarrow{\textbf{A}}/\theta^{\prime}$.

###### Proof.

Suppose first that there is a homomorphism
$h:{\textbf{A}}/\theta\longrightarrow{\textbf{A}}/\theta^{\prime}$. Since
${\textbf{A}}\cong{\textbf{A}}/\theta\times{\textbf{A}}/\theta^{\prime}$ for
$u\in A$, $u=(a/\theta,b/\theta^{\prime})$, we set $f(u)=a/\theta$; then
$f:{\textbf{A}}\longrightarrow{\textbf{A}}/\theta$ is clearly an epimorphism,
since $(a/\theta,a/\theta^{\prime})\in A$ for all $a\in A$. Let

$g(a/\theta)=(a/\theta,h(a/\theta)).$

One can check that $g$ is a homomorphism with standard calculations and
clearly $fg=id_{{\textbf{A}}/\theta}$. Hence ${\textbf{A}}/\theta$ is a
retract of A.

Conversely suppose that $f,g$ witness a retraction from ${\textbf{A}}/\theta$
in A; then if $g(a/\theta)=(u/\theta,v/\theta^{\prime})$, set
$h(a/\theta)=v/\theta^{\prime}$. It is then easy to see that $h$ is a
homomorphism and the thesis holds. ∎

Observe that in any $\mathsf{FL}$-algebra, every compact (i.e., finitely
generated) congruence is principal; as a matter of fact if A is in
$\mathsf{FL}$, $X=\\{(a_{1},b_{1}),\dots,(a_{n},b_{n})\\}$ is a finite set of
pairs from $A$ and $p=\bigwedge_{i=1}^{n}[(a_{i}\backslash
b_{i})\wedge(b_{i}\backslash a_{i})\wedge 1]$ then
$\vartheta_{\textbf{A}}(X)=\vartheta_{\textbf{A}}(p,1)$.

######  Theorem 5.16.

Let $\mathsf{Q}$ be a quasivariety of $\mathsf{FL}_{w}$-algebras in which
every principal congruence is a factor congruence; then $\mathsf{Q}$ has
projective unifiers.

###### Proof.

Let ${\textbf{F}}_{\mathsf{Q}}(X)/\theta$ be a finitely presented unifiable
algebra in $\mathsf{Q}$; then there is an onto homomorphism from
${\textbf{F}}_{\mathsf{Q}}(X)/\theta(\Sigma)$ to
${\textbf{F}}_{\mathsf{Q}}={\textbf{2}}$. Now $\theta=\theta(\Sigma)$ is a
principal congruence, hence it is a factor congruence with witness
$\theta^{\prime}$, i.e.
${\textbf{F}}_{\mathsf{Q}}(X)\cong{\textbf{F}}_{\mathsf{Q}}(X)/\theta\times{\textbf{F}}_{\mathsf{Q}}(X)/\theta^{\prime}$.
If $\theta^{\prime}=1_{\textbf{A}}$, then
${\textbf{F}}_{\mathsf{Q}}(X)={\textbf{F}}_{\mathsf{Q}}(X)/\theta$ and so it
is projective. Otherwise ${\textbf{F}}_{\mathsf{Q}}={\textbf{2}}$ is
embeddable in ${\textbf{F}}_{\mathsf{Q}}(X)/\theta^{\prime}$; hence there is a
homomorphism from from ${\textbf{F}}_{\mathsf{Q}}(X)/\theta$ to
${\textbf{F}}_{\mathsf{Q}}(X)/\theta^{\prime}$. By Lemma 5.15
${\textbf{F}}_{\mathsf{Q}}(X)/\theta$ is a retract of
${\textbf{F}}_{\mathsf{Q}}(X)$, i.e. it is projective. ∎

So any quasivariety of $\mathsf{FL}_{w}$-algebras with the property that every
principal congruence is a factor congruence is active universally complete
(Theorem 3.22); really it is active primitive universally complete, because
${\textbf{F}}_{\mathsf{Q}}$ is the two-element algebra for any quasivariety
$\mathsf{Q}$ of $\mathsf{FL}_{w}$-algebras (Theorem 3.26). We observe in
passing that for any $\mathsf{FL}_{w}$ algebra every factor congruence is
principal; this is because every variety of $\mathsf{FL}_{w}$-algebras is
Kollár and congruence distributive. Discriminator varieties of
$\mathsf{FL}_{ew}$-algebras have been completely described in [60]; as a
consequence we have:

######  Theorem 5.17.

For a variety $\mathsf{V}$ of $\mathsf{FL}_{ew}$-algebras the following are
equivalent:

  1. (1)

$\mathsf{V}$ is a discriminator variety;

  2. (2)

$\mathsf{V}$ is semisimple, i.e. all the subdirectly irreducible members of
$\mathsf{V}$ are simple;

  3. (3)

there is an $n\in\mathbb{N}$ such that $\mathsf{V}\vDash x\vee\neg
x^{n}\approx 1$;

  4. (4)

for any ${\textbf{A}}\in\mathsf{V}$ every compact (i.e. principal) congruence
is a factor congruence.

###### Proof.

The equivalence of (1), (2) and (3) has been proved in [60]. Assume then (1);
it is well-known that in every discriminator variety every principal
congruence is a factor congruence. In fact if $\mathsf{V}$ is a discriminator
variety with discriminator term $t(x,y,z)$ let for any
${\textbf{A}}\in\mathsf{V}$ and $a,b\in A$

$\displaystyle\theta_{\textbf{A}}(a,b)=\\{(u,v):t(a,b,u)=t(a,b,v)\\}$
$\displaystyle\gamma_{\textbf{A}}(a,b)=\\{(u,v):t(a,t(a,b,u),u)=t(a,t(a,b,v),v)\\}.$

Using the properties of the discriminator term it is easy to verify that they
are congruences and the complement of each other; since discriminator
varieties are congruence permutable they are factor congruences and (4) holds.

Conversely assume (4) and let A be a subdirectly irreducible member of
$\mathsf{V}$. Let $\mu_{\textbf{A}}$ be the minimal nontrivial congruence of
A; then $\mu_{\textbf{A}}$ is principal, so it must be a factor congruence.
This is possible if and only if $\mu_{\textbf{A}}=1_{\textbf{A}}$; therefore A
is simple, and $\mathsf{V}$ is semisimple. ∎

######  Corollary 5.18.

Every discriminator (or, equivalently, semisimple) variety $\mathsf{V}$ of
$\mathsf{FL}_{ew}$-algebras is active primitive universal.

We observe that Theorem 5.16 does not add anything as far as
$\mathsf{BL}$-algebras are concerned; in fact any discriminator variety of
$\mathsf{FL}_{ew}$-algebras must satisfy $x^{n}\approx x^{n+1}$ for some $n$
[60] and the varieties of $\mathsf{BL}$-algebras with that property are
exactly the locally finite varieties, which we already pointed out are active
universally complete.

####  5.2.2. Passive structural completeness

A particularly interesting application of our characterization of passive
structurally complete varieties is in the subvariety of integral and
$0$-bounded FL-algebras. Let us rephrase Theorem 4.23 in this setting. First,
using residuation it is easy to see that every finite set of identities in
$\mathsf{FL}$ is equivalent to a single identity. Moreover, in every
subquasivariety $\mathsf{Q}$ of $\mathsf{FL}_{w}$, the smallest free algebra
${\textbf{F}}_{\mathsf{Q}}$ is the 2-element Boolean algebra 2, and its
generated quasivariety is the variety of Boolean algebras.

######  Corollary 5.19.

Let $\mathsf{Q}$ be a quasivariety of $\mathsf{FL}_{w}$-algebras, then the
following are equivalent:

  1. (1)

$\mathsf{Q}$ is passive structurally complete;

  2. (2)

every trivializing identity in the variety of Boolean algebras is trivializing
in $\mathsf{Q}$;

  3. (3)

every nontrivial finitely presented algebra is unifiable.

  4. (4)

every nontrivial algebra is unifiable.

The previous corollary has a possibly more transparent shape from the point of
view of the logics. Let us call a formula $\varphi$ in the language of
$\mathsf{FL}$-algebras _explosive in a logic $\mathcal{L}$_, with consequence
relation $\vdash_{\mathcal{L}}$, if $\varphi\vdash_{\mathcal{L}}\delta$ for
all formulas $\delta$ in the language of $\mathcal{L}$. Moreover, we call
$\varphi$ a _contradiction in $\mathcal{L}$_ if
$\varphi\vdash_{\mathcal{L}}0$. Since $\mathsf{FL}_{w}$-algebras are
$0$-bounded, it is clear that contradictions coincide with explosive formulas
in all axiomatic extensions of $\mathcal{F}\mathcal{L}_{w}$.

######  Corollary 5.20.

Let $\mathcal{L}$ be an axiomatic extension of $\mathcal{FL}_{w}$, then the
following are equivalent:

  1. (1)

$\mathcal{L}$ is passively structurally complete.

  2. (2)

Every contradiction of classical logic is explosive in $\mathcal{L}$.

  3. (3)

Every passive rule of $\mathcal{L}$ has explosive premises.

Let us first explore the consequences of the equivalence between (1) and (2)
in Corollary 5.20. It is well known that intuitionistic logic is passively
structurally complete (reported by Wronski at the 51st Conference on the
History of Logic, Krakow, 2005). This is easily seen by Corollary 5.20,
indeed, observe that any contradiction of classical logic $\varphi$ is such
that its negation $\neg\varphi$ is a theorem of classical logic. Using the
Glivenko translation and the deduction theorem, we obtain that $\varphi$ is
explosive in intuitionistic logic as well, which is then passively
structurally complete. We will now show how this argument can be extended to a
wide class of logics.

Let us write the negations corresponding to the two divisions as $\neg
x=x\backslash 0$ and $\sim x=0/x$. Following [44, 45], we say that two logics
$\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ are _Glivenko equivalent_ if for all
formuals $\varphi$:

$\vdash_{\mathcal{L}_{1}}\neg\varphi\;\;\mbox{ iff
}\;\;\vdash_{\mathcal{L}_{2}}\neg\varphi$

(equivalently, $\vdash_{\mathcal{L}_{1}}\sim\varphi\mbox{ iff
}\vdash_{\mathcal{L}_{2}}\sim\varphi$). Given a logic $\mathcal{L}$, we call
_Glivenko logic of $\mathcal{L}$_ the smallest substructural logic that is
Glivenko equivalent to $\mathcal{L}$ . Moreover, we call _Glivenko logic of
$\mathcal{L}$ with respect to $\mathcal{L}^{\prime}$_, and denote it with
$\mathcal{G}_{\mathcal{L}^{\prime}}(\mathcal{L})$ the smallest extension of
$\mathcal{L}^{\prime}$ that is Glivenko equivalent to $\mathcal{L}$ (all these
notions make sense by the results in [44, 45]).
$\mathcal{G}_{\mathcal{L}^{\prime}}(\mathcal{L})$ is axiomatized relatively to
$\mathcal{L}^{\prime}$ by the set of axioms
$\\{\neg\sim\varphi:\vdash_{\mathcal{L}}\varphi\\}$, or equivalently by the
set $\\{\sim\neg\varphi:\vdash_{\mathcal{L}}\varphi\\}$.

Here we are interested in the Glivenko equivalent of classical logic with
respect to $\mathcal{FL}_{w}$. From the algebraic perspective, this
corresponds to the largest subvariety of $\mathsf{FL}_{w}$ that is Glivenko
equivalent to Boolean algebras, $\mathsf{G}_{\mathsf{FL}_{w}}(\mathsf{B})$.
The latter is axiomatized in [43, Corollary 8.33] as the subvariety of
$\mathsf{FL}_{w}$ satisfying:

  1. (1)

$\sim(x\land y)=\sim(xy)$

  2. (2)

$\sim(x\backslash y)=\sim(\neg x\lor y)$

  3. (3)

$\neg(x\backslash y)=\neg(\sim x\lor y)$

  4. (4)

$\sim(x\backslash y)=\sim(\neg\sim x\backslash\neg\sim y)$

  5. (5)

$\sim(x/y)=\sim(\neg\sim x/\neg\sim y)$.

######  Theorem 5.21.

Every axiomatic extension $\mathcal{L}$ of the Glivenko logic of classical
logic with respect to $\mathcal{FL}_{w}$ is passively structurally complete.

###### Proof.

Consider a contradiction of classical logic $\varphi$, by the deduction
theorem $\vdash_{\mathcal{C}\mathcal{L}}\neg\varphi$ (where
$\vdash_{\mathcal{C}\mathcal{L}}$ is the consequence relation of classical
logic). Since $\mathcal{L}$ is Glivenko equivalent to classical logic,
$\vdash_{\mathcal{L}}\neg\varphi$. It can be easily checked that this implies
that $\varphi\vdash_{\mathcal{L}}0$ (it is a consequence of the parametrized
local deduction theorem which holds in every extension of $\mathcal{FL}$ [43],
but it is also straightforward to see in models). Thus $\varphi$ is a
contradiction of $\mathcal{L}$, or equivalently it is explosive in
$\mathcal{L}$, which is then passively structurally complete by Corollary
5.20. ∎

Thus, every subvariety of $\mathsf{G}_{\mathsf{FL}_{w}}(\mathsf{B})$ is
passive structurally complete. In particular, the commutative subvariety
$\mathsf{G}_{\mathsf{FL}_{ew}}(\mathsf{B})$ is the variety of
pseudocomplemented $\mathsf{FL}_{ew}$-algebras ([39]), axiomatized by

$x\land\neg x\approx 0.$

Examples of passive structurally complete varieties then include Heyting
algebras, Stonean MTL-algebras and as a consequence, e.g., product algebras
and Gödel algebras.

We observe that these are not all of the passive structural complete varieties
of $\mathsf{FL}_{w}$ (nor of $\mathsf{FL}_{ew}$). Let us indeed obtain a
different kind of examples.

######  Definition 5.22.

We say that a variety $\mathsf{V}$ has a _Boolean retraction term_ if there
exists a term $t$ in the language of residuated lattices (i.e., $0$-free) such
that, for every ${\textbf{A}}\in\mathsf{V}$, $t$ defines an idempotent
endomorphism on A whose image is the Boolean skeleton of A, that is, the set
of complemented elements of A.

Varieties with a Boolean retraction term have been studied at length by
Cignoli and Torrens in a series of papers, see in particular [28]. These are
all varieties in which all nontrivial algebras retract onto a nontrivial
Boolean algebra, thus they satisfy the hypotheses of Corollary 5.19 and they
are passive structurally complete. Some of these varieties have been shown in
[7] to have projective unifiers, thus they satisfy Theorem 3.16 and they are
non-negative universally complete. Among those we cite some varieties of
interest in the realm of many-valued logics: the variety of product algebras,
the variety generated by perfect MV-algebras, the variety $\mathsf{NM^{-}}$ of
nilpotent minimum algebras without negation fixpoint and some varieties that
have been called nilpotent product in [6] or [5].

We will see that in the representable variety of $\mathsf{FL}_{ew}$,
$\mathsf{MTL}$, we can fully characterize passive structurally complete
varieties as those with a Boolean retraction term. By [28], the largest
subvariety of $\mathsf{M}\mathsf{T}\mathsf{L}$ with a Boolean retraction term
is axiomatized relatively to $\mathsf{M}\mathsf{T}\mathsf{L}$ by the Di Nola-
Lettieri equation:

(DL) $(x+x)^{2}=x^{2}+x^{2}$

where $x+y=\neg(\neg x\cdot\neg y)$. The latter identity has been introduced
by Di Nola and Lettieri to axiomatize within MV-algebras the variety generated
by the Chang algebra. This variety is called $\mathsf{sDL}$ in [73]
($\mathsf{BP}_{0}$ in [67, 10]), and it includes, for instance:
pseudocomplemented MTL-algebras (also called SMTL-algebras), and thus Gödel
algebras and product algebras; involutive $\mathsf{BP}_{0}$-algebras and thus
the variety generated by perfect MV-algebras and nilpotent minimum algebras
without negation fixpoint.

Let us say that an element of an $\mathsf{FL}_{ew}$-algebra A has finite order
$n$ if $x^{n}=0$, and infinite order if there is no such $n$. We call
_perfect_ an algebra ${\textbf{A}}\in\mathsf{FL}_{ew}$ such that, for all
$a\in A$, $a$ has finite order if and only if $\neg a$ has infinite order.
Now, $\mathsf{sDL}$ turns out to be the variety generated by the perfect
chains (see [73, 10]).

######  Lemma 5.23.

A chain ${\textbf{A}}\in\mathsf{FL}_{ew}$ is perfect if and only if there is
no element with finite order $a\in A$ such that $a\geq\neg a$.

###### Proof.

By order preservation, if there is an element $a\in A,a\geq\neg a,a^{n}=0$,
then both $a$ and its negation have finite order, thus the chain is not
perfect. Suppose now a chain A is not perfect. Observing that for every
element $x\in A$ it cannot be that both $x$ and $\neg x$ gave infinite order,
we get that there is an element $a\in A$ such that both $a$ and its negation
$\neg a$ have finite order. If $a\not\geq\neg a$, since A is a chain, $a<\neg
a$. Then $\neg\neg a\leq\neg a$, and they both have finite order. ∎

######  Theorem 5.24.

For a subvariety $\mathsf{V}$ of $\mathsf{MTL}$ the following are equivalent:

  1. (1)

$\mathsf{V}$ is passive structurally complete;

  2. (2)

$\mathsf{V}$ is a subvariety of $\mathsf{sDL}$.

###### Proof.

Since subvarieties of $\mathsf{sDL}$ have a Boolean retraction term (2)
implies (1) by Corollary 5.19. Suppose now that
$\mathsf{V}\nsubseteq\mathsf{sDL}$. Then there is a chain A in $\mathsf{V}$
that is not perfect. By Lemma 5.23, there exists $a\in A,a\geq\neg
a,a^{n}=0\mbox{ for some }n\in\mathbb{N}.$ Thus, $\neg(a\lor\neg a)^{n}=1$.
But the identity $\neg(x\lor\neg x)^{n}=0$ holds in Boolean algebras. Thus
$\neg(x\lor\neg x)^{n}\approx 1$ is trivializing in Boolean algebras but not
in $\mathsf{V}$. By Corollary 5.19, $\mathsf{V}$ is not passive structurally
complete and thus (1) implies (2). ∎

Notice that the previous theorem also implies that a variety of MTL-algebras
that is not a subvariety of $\mathsf{sDL}$ cannot be structurally complete.

We mention that structural completeness in subvarieties of $\mathsf{MTL}$ (or
their logical counterparts) has been studied by several authors: e.g., [76]
and [49] for Łukasiewicz logics, [36] Gödel logic, and [29] for fuzzy logics
in the MTL framework; in the latter the authors show for instance that all
subvarieties of pseudocomplemented MTL-algebras ($\mathsf{SMTL}$) are passive
structurally complete. This result is here obtained as a consequence of
Theorem 5.24, since $\mathsf{SMTL}$ is a subvariety of $\mathsf{sDL}$. From
the results mentioned above and the characterzation theorem, it also follows
that the only varieties of MV-algebras (the equivalent algebraic semantics of
infinite-valued Łukasiewicz logic) that are structurally complete are Boolean
algebras and the variety generated by perfect MV-algebras (this result has
been obtained following a different path in [49]).

We also remark that a variety of $\mathsf{FL}_{ew}$-algebras can be at most
non-negative universally complete since trivial algebras are finitely
presented and not unifiable (unifiability is a necessary condition for
universal completeness by Theorem 3.4); by Proposition 4.20 this happens if
and only if the variety is active universally complete and passive
structurally complete. Thus, for instance, a semisimple variety of
$\mathsf{FL}_{ew}$-algebras satisfying the conditions in Corollary 5.19 would
be non-negative universally complete. We stress that this observation is not
of particular interest in MTL-algebras, since the only discriminator variety
in $\mathsf{sDL}$ is the variety of Boolean algebras. Indeed, consider a chain
A in a discriminator variety $\mathsf{V}$ in $\mathsf{sDL}$. Then there is
some $n\in\mathbb{N}$ such that $\mathsf{V}\models x\lor\neg x^{n}\approx 1$.
Let now $a\in A$; either $a$ has finite order, and then from $a\lor\neg a^{n}$
we obtain that $a=1$, or $a$ has infinite order, and then $\neg a$ has finite
order. So by the analogous reasoning $\neg a=1$. Therefore A is the two-
element chain, and $\mathsf{V}$ is the variety of Boolean algebras.

##  6\. Conclusions

In Figure 8 we display several classes of varieties that we have considered in
this paper (and the labels should be self explanatory); we are dropping the
hereditary subclasses to avoid clutter. Observe that this is really a meet
semilattice under inclusion.

$U$$NNU$$AU$$S$$S+PU$$PU$$PS$$AS$ Figure 8. The universal and structural
completeness classes.

Almost all the classes are provably distinct.

  1. (1)

The variety of bounded distributive lattice is structurally complete
(Proposition 5.5) but it is neither passive universally complete, since it is
Kollár and the least free algebra is not trivial, nor non-negative universally
complete (Proposition 5.4). Hence $S\neq NNU,S+PU$.

  2. (2)

The variety of Boolean algebras is non-negative universally complete but not
universally complete (Example 3.18) so $NNU\neq U$.

  3. (3)

Any locally finite variety of $\mathsf{BL}$-algebras is active universally
complete and some of them are not non-negative universally complete (Example
3.24), so $AU\neq NNU$.

  4. (4)

The variety in Example 4.19 is active structurally complete but not active
universally complete, hence $AS\neq AU$.

  5. (5)

Any locally finite variety of bounded semidistributive lattices different from
the distributive variety is passive structurally complete (Corollary 5.12) but
not structurally complete, since the only structurally complete variety of
bounded distributive lattices is the distributive variety (Proposition 5.5);
as above it is also not passive universally complete. Hence $PS\neq S,PU$.

  6. (6)

The variety ${\mathbf{V}}({\textbf{M}}_{3,3}^{+})$ (Section 5.1) is passive
universally complete, as any variety of lattices, but it is not structurally
complete since
${\mathbf{Q}}({\textbf{M}}_{3,3}^{+})\nsubseteq{\mathbf{V}}({\textbf{M}}_{3,3}^{+})$;
hence $PU\neq S+PU$.

  7. (7)

Example 7.11 in [35] shows that $AS\neq S$.

Moreover for the primitive counterparts:

  1. (1)

the variety ${\mathbf{V}}({\textbf{F}})$ generated by the Fano lattice is
structurally complete and passive universally complete but not primitive
(Section 5.1).

  2. (2)

the variety of De Morgan lattices (Example 3.29) is active universally
complete but not active primitive universal.

  3. (3)

the variety of injective monounary algebras is active structurally complete
but not active primitive structural (Example 7.2 in [35]).

There are three examples that we were not able to find, which would guarantee
total separation of all the classes we have considered:

  1. (1)

A (quasi)variety that is structurally complete and passive universally
complete, but not universally complete.

  2. (2)

A non-negative universally complete (quasi)variety such that not all
subquasivarieties are non-negative universally complete.

  3. (3)

A universally complete variety which is not primitive universal.

The natural example for (3) would be a locally finite variety with exact
unifiers having a subvariety without exact unifiers. However we are stuck
because of lack of examples: we have only one unifiable locally finite variety
with exact (non projective) unifiers, i.e. the variety of distributive
lattices, which is trivially primitive universal. A similar situation happens
for (2); all the examples of non-negative universally complete varieties we
have are either equationally complete and congruence distributive (so they do
not have nontrivial subquasivarieties), or else are active universally
complete just by consequence of their characterization (such as the
subvarieties of $\mathsf{FL}_{ew}$ in Section 5.2). Then we have Stone
algebras that are not equationally complete but the only nontrivial subvariety
is the variety of Boolean algebras, that is non-negative universally complete.
Now from Corollary 2.30 it is immediate that every subquasivariety of
$\mathsf{ST}$ is non-negative universally complete. In conclusion a deeper
investigation of universally complete and non-negative universally complete
varieties is needed.

For (1) the situation is (slightly) easier to tackle: any primitive variety of
lattices that is not universally complete gives a counterexample. While it
seems impossible that all the primitive varieties in Section 5.1.1 are
universally complete, actually proving that one it is not does not seem easy.
This is due basically to the lack of information on free algebras in specific
varieties of lattices, such as for instance ${\mathbf{V}}({\textbf{M}}_{3})$;
note that this variety is locally finite and hence all the finitely generated
free algebras are finite. But we are not aware of any characterization.

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