Proof The assumption that $\vert F \vert \leqslant 1$ for every $\langle \boldsymbol {A}, F \rangle \in \mathsf {M}$ implies that $x, y, \varphi (x, \vec {z}) \vdash \varphi (y, \vec {z})$ for every formula $\varphi (x, \vec {z}) \in Fm(\vdash )$ . By Proposition 3.8(i) we conclude that $\vdash $ is truth-minimal.

Proof For every $j \in I$ , we denote the natural projection map by $\pi _{j} \colon \boldsymbol {A} \to \boldsymbol {A}_{j}$ , and set $G_{j} := \pi _{j}[G]$ .

Then we turn to prove that $G = A \cap \prod _{i \in I}G_{i}$ . The inclusion from left to right is clear. To prove the other one, consider an element $a \in A \cap \prod _{i \in I}G_{i}$ . For every $j \in I$ there is $c_{j} \in G$ such that $c_{j}(j) = a(j)$ . Then consider the basic operation $x \multimap _{j} y$ of $\bigotimes _{i \in I}{\vdash _{i}}$ , whose j-th component is the projection on the first coordinate, and whose i-th component is the projection on the second coordinate for every $i \in I \smallsetminus \{ j \}$ . Since $c_{j}(j) = a(j)$ ,

(5) $$ \begin{align} a \multimap_{j}^{\boldsymbol{A}} c_{j} = c_{j} \in G. \end{align} $$

Now, consider some distinct variables $\{ y_{j} \colon j \in I \} \cup \{ x \} \subseteq Fm(\bigotimes _{i \in I}{\vdash _{i}})$ . Bearing in mind that $\bigotimes _{i \in I}{\vdash _{i}}$ is the logic induced by the class $\bigotimes _{i \in I}\mathsf {Mod}^{\equiv }(\vdash _{i})$ , it is easy to see that the following rule is valid in $\bigotimes _{i \in I}{\vdash _{i}}$ :

$$ \begin{align*} \{ x \multimap_{j} y_{j} \colon j \in I \} \rhd x. \end{align*} $$

Together with (5) and the fact that G is a deductive filter of $\bigotimes _{i \in I}{\vdash _{i}}$ on $\boldsymbol {A}$ , this implies $a \in G$ . Hence we conclude

(6) $$ \begin{align} G = A \cap \prod_{i \in I}G_{i}. \end{align} $$

Consider an index $j \in I$ . To conclude the proof, it suffices to show that $G_{j}$ is a filter of $\vdash _{j}$ on $\boldsymbol {A}_{j}$ . To prove this, suppose that $\varGamma \vdash _{j} \varphi $ and consider a homomorphism $h \colon \boldsymbol {Fm}(\vdash _{j}) \to \boldsymbol {A}_{j}$ such that $h[\varGamma ] \subseteq G_{j}$ . For every n-ary term $\psi (z_{1}, \dots , z_{n}) \in Fm(\vdash _{j})$ , let $\hat {\psi }(z_{1}, \dots , z_{n})$ be an arbitrary basic n-ary operation of $\bigotimes _{i \in I}{\vdash _{i}}$ whose j-th component is $\psi $ . Moreover, consider a variable $y \in Fm(\bigotimes _{i \in I}{\vdash _{i}})$ not occurring in $\varGamma $ .Footnote ² Again bearing in mind that $\bigotimes _{i \in I}{\vdash _{i}}$ is the logic induced by the class $\bigotimes _{i \in I}\mathsf {Mod}^{\equiv }(\vdash _{i})$ , it is clear that the following rule is valid in $\bigotimes _{i \in I}{\vdash _{i}}$ :

(7) $$ \begin{align} \{ \hat{\gamma} \multimap_{j} y \colon \gamma \in \varGamma \} \rhd \hat{\varphi} \multimap_{j} y. \end{align} $$

Now, choose an element $a \in G$ , and let $f \colon \boldsymbol {Fm}(\bigotimes _{i \in I}{\vdash _{i}}) \to \boldsymbol {A}$ be a homomorphism such that $f(y) = a$ and $f(x)(j) = h(x)$ for every variable $x \in Fm(\vdash _{j})$ . Observe that there exists such an f, since the projection $\pi _{j} \colon A \to A_{j}$ is surjective. For every $i \in I \smallsetminus \{ j \}$ and $\gamma \in \varGamma $ , we have

$$ \begin{align*} f(\hat{\gamma} \multimap_{j} y)(i) &= a(i) \in \pi_{i}[G] = G_{i}\\ f(\hat{\gamma} \multimap_{j} y)(j) &= h(\gamma)\in G_{j}. \end{align*} $$

Together with (6), this implies

$$ \begin{align*} f(\hat{\gamma} \multimap_{j} y) \in A \cap \prod_{i \in I}G_{i} = G \text{, for every }\gamma \in \varGamma. \end{align*} $$

The above display, together with (7) and the fact that G is a deductive filter of $\bigotimes _{i \in I}{\vdash _{i}}$ on $\boldsymbol {A}$ , yields that $f(\hat {\varphi } \multimap _{j}y) \in G$ . As a consequence, we obtain

$$ \begin{align*} h(\varphi) = f(\hat{\varphi} \multimap_{j}y)(j) \in \pi_{j}[G] = G_{j}. \end{align*} $$

Hence we conclude that $G_{j}$ is a deductive filter of $\vdash _{j}$ on $\boldsymbol {A}_{j}$ .

Proof It is straightforward that the class of truth-minimal logics is closed under term-equivalence and compatible expansions. In the light of Theorem 2.2, it only remains to prove that this class is closed under the formation of non-indexed products of sets.

To prove this, consider a family $\{ {\vdash _{i}} \colon i \in I \}$ of truth-minimal logics. Moreover, consider $\langle \boldsymbol {A}, F \rangle , \langle \boldsymbol {A}, G \rangle \in \mathsf {Mod}^{\equiv }(\bigotimes _{i \in I} {\vdash _{i}})$ with $\emptyset \ne G \subseteq F$ . We need to show that $F = G$ . By [Reference Jansana and Moraschini24, Proposition 4.5] there is a family of matrices $\{ \langle \boldsymbol {A}_{i}, F_{i}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{i}) \colon i \in I \}$ such that $\langle \boldsymbol {A}, F \rangle \leqslant _{\text {sd}} \bigotimes _{i \in I}\langle \boldsymbol {A}_{i}, F_{i}\rangle $ . By Lemma 5.3, for every $j \in I$ there is a deductive filter $G_{j}$ of $\vdash _{j}$ on $\boldsymbol {A}_{j}$ such that

(8) $$ \begin{align} \emptyset \ne G_{j} \subseteq F_{j} \end{align} $$

and $G = A \cap \prod _{i \in I}G_{i}$ . Now, from the fact that $G_{j}$ is a deductive filter of $\vdash _{j}$ and $G_{j} \subseteq F_{j}$ it follows that . Therefore, bearing in mind that is the identity relation, we conclude that the same holds for . In particular, this yields

(9) $$ \begin{align} \langle \boldsymbol{A}_{j}, G_{j}\rangle \in \mathsf{Mod}^{\equiv}(\vdash_{j}). \end{align} $$

Finally, for every $j \in J$ we can apply the fact that $\vdash _{j}$ is truth-minimal to (8, 9), obtaining that $F_{j} = G_{j}$ . As a consequence, we get

$$ \begin{align*} G = A \cap \prod_{i \in I}G_{i} = A \cap \prod_{i \in I}F_{j} = F. \end{align*} $$

Hence we conclude that $\bigotimes _{i \in I}{\vdash _{i}}$ is truth-minimal.

Proof From [Reference Jansana and Moraschini24, Proposition 2.2(ii)] we obtain . On the other hand, since $\langle \boldsymbol {A}, A \rangle \in \mathsf {Mod}^{\equiv }(\vdash )$ , the congruence is the identity relation. But the fact that $A \times A$ is the identity relation implies that A is a singleton, as desired.

Proof By Lemma 5.6 for every $i = 1, 2$ there are $\langle \boldsymbol {C}_{i}, F^{\dagger }_{i} \rangle , \langle \boldsymbol {C}_{i}, G^{\dagger }_{i} \rangle \in \mathsf {Mod}^{\equiv }(\vdash _{i})$ such that $\emptyset \ne G^{\dagger }_{i} \subsetneq F^{\dagger }_{i} \subsetneq C_{i}$ and $\omega \leqslant \vert C_{i} \vert = \vert G^{\dagger }_{i} \vert = \vert F^{\dagger }_{i} \smallsetminus G^{\dagger }_{i} \vert = \vert C_{i} \smallsetminus F^{\dagger }_{i} \vert $ .

Let then $\kappa := \max \{ \vert C_{1} \vert , \vert C_{2} \vert \}$ . Consider also $i = 1, 2$ . Since $\mathsf {Mod}^{\equiv }(\vdash _{i})$ is closed under direct powers [Reference Jansana and Moraschini24, Lemma 2.3], we obtain

(13) $$ \begin{align} \langle \boldsymbol{C}_{i}^{\kappa}, (F^{\dagger}_{i})^{\kappa} \rangle, \langle \boldsymbol{C}_{i}^{\kappa}, (G^{\dagger}_{i})^{\kappa} \rangle \in \mathsf{Mod}^{\equiv}(\vdash_{i}). \end{align} $$

Moreover, since $\emptyset \ne G_{i}^{\dagger } \subsetneq F^{\dagger }_{i} \subsetneq C_{i}$ , we have

$$ \begin{align*} \emptyset\ne G_{i}^{\dagger\kappa} \subsetneq F^{\dagger\kappa}_{i} \subsetneq C_{i}^{\kappa}. \end{align*} $$

Finally, since $\omega \leqslant \vert C_{i} \vert = \vert G^{\dagger }_{i} \vert = \vert F^{\dagger }_{i} \smallsetminus G^{\dagger }_{i} \vert = \vert C_{i} \smallsetminus F^{\dagger }_{i} \vert \leqslant \kappa $ , we have

(14) $$ \begin{align} \kappa = \vert C_{i}^{\kappa} \vert = \vert G^{\dagger\kappa}_{i} \vert = \vert F^{\dagger\kappa}_{i} \smallsetminus G^{\dagger\kappa}_{i} \vert = \vert C_{i}^{\kappa} \smallsetminus F^{\dagger\kappa}_{i} \vert. \end{align} $$

Then consider some sets $\emptyset \ne G \subsetneq F \subsetneq B$ such that

$$ \begin{align*} \kappa = \vert B \vert = \vert G \vert = \vert F \smallsetminus G \vert = \vert B \smallsetminus F \vert. \end{align*} $$

From (13) and (14) it easily follows that for each $i = 1, 2$ there is an $\mathscr {L}_{\vdash _{i}}$ -algebra $\boldsymbol {B}_{i}$ with universe B such that $\langle \boldsymbol {B}_{i}, F \rangle \cong \langle \boldsymbol {C}_{i}^{\kappa }, (F^{\dagger }_{i})^{\kappa } \rangle $ , $\langle \boldsymbol {B}_{i}, G\rangle \cong \langle \boldsymbol {C}_{i}^{\kappa }, (G^{\dagger }_{i})^{\kappa } \rangle $ and, therefore, $\langle \boldsymbol {B}_{i}, F \rangle , \langle \boldsymbol {B}_{i}, G\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{i})$ .

Proof Consider two logics $\vdash _{1}$ and $\vdash _{2}$ that are not truth-minimal. It will be enough to construct a logic $\vdash $ that is not truth-minimal and in which $\vdash _{1}$ and $\vdash _{2}$ are interpretable.

To this end, observe that for every $i = 1, 2$ there are $\langle \boldsymbol {A}_{i}, F_{i}\rangle , \langle \boldsymbol {A}_{i}, G_{i}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{i})$ such that $\emptyset \ne G_{i} \subsetneq F_{i}$ . From Lemma 5.7 we obtain

$$ \begin{align*} \langle \boldsymbol{B}_{1}, F \rangle, \langle \boldsymbol{B}_{1}, G \rangle \in \mathsf{Mod}^{\equiv}(\vdash_{1}) \text{ and }\langle \boldsymbol{B}_{2}, F \rangle, \langle \boldsymbol{B}_{2}, G \rangle \in \mathsf{Mod}^{\equiv}(\vdash_{2}) \end{align*} $$

for some sets $\emptyset \ne G \subsetneq F \subsetneq B$ , an $\mathscr {L}_{\vdash _{1}}$ -algebra $\boldsymbol {B}_{1}$ , and an $\mathscr {L}_{\vdash _{2}}$ -algebra $\boldsymbol {B}_{2}$ such that $B_{1} = B_{2} = B$ .

Let $\boldsymbol {B}$ be the common expansion of $\boldsymbol {B}_{1}$ and $\boldsymbol {B}_{2}$ with all finitary operations on B. Moreover, let $\vdash $ be the logic, formulated in $\kappa := \max \{ \vert Fm(\vdash _{1}) \vert , \vert Fm(\vdash _{2}) \vert \}$ variables, induced by the set of matrices

$$ \begin{align*} \mathsf{K} := \{ \langle \boldsymbol{B}, F \rangle, \langle \boldsymbol{B}, G \rangle\}. \end{align*} $$

We claim that $\vdash $ is equivalential. To prove this, consider elements $0 \in B \smallsetminus F$ and $1 \in G$ (this is possible, since $\emptyset \ne G$ and $F\subsetneq B$ ). Then let $x \multimap y$ be the binary basic operation of $\boldsymbol {B}$ defined as follows for every $a, c \in B$ ,

$$ \begin{align*} a \multimap^{\boldsymbol{B}} c := \left\{\begin{array}{@{\,}ll} 1 & \text{if}\ a=c,\\ 0 & \text{otherwise.}\\ \end{array} \right. \end{align*} $$

Bearing in mind that $0 \notin F \cup G$ and $1 \in F \cap G$ , it is not hard to see that the set $\varDelta (x, y) := \{ x \multimap y \}$ satisfies the conditions in the right hand side of [Reference Jansana and Moraschini24, Theorem 2.7]. As a consequence we obtain that $\vdash $ is equivalential, establishing the claim.

Since $\boldsymbol {B}$ is endowed with all unary constant maps, we have $\mathbb {S}(\mathsf {K}) = \mathsf {K}$ . Moreover, for each pair of different $a, c \in B$ , we have

$$ \begin{align*} a \multimap^{\boldsymbol{B}} c = 0 \notin G \cup F \text{ and }a \multimap^{\boldsymbol{B}} a = 1 \in G \cap F. \end{align*} $$

By [Reference Jansana and Moraschini24, Proposition 2.2(i)] this implies that the matrices in $\mathsf {K}$ are reduced. This fact, together with the claim and $\mathbb {S}(\mathsf {K}) = \mathsf {K}$ , allows us to apply [Reference Jansana and Moraschini24, Proposition 3.9] obtaining that the identity maps are interpretations of $\vdash _{1}$ and $\vdash _{2}$ into $\vdash $ . Moreover, $\vdash $ is not truth-minimal, as witnessed by the fact that $\mathsf {K} \subseteq \mathsf {Mod}^{\equiv }(\vdash )$ and $\emptyset \ne G \subsetneq F$ .

Proof See [Reference Raftery32, Proposition 17 and Theorem 2.8].

Proof Consider two logics $\vdash _{1}$ and $\vdash _{2}$ with theorems that are not truth-equational. It will be enough to construct a logic $\vdash $ that is not truth-equational and in which $\vdash _{1}$ and $\vdash _{2}$ are interpretable.

First we claim that for every $i = 1, 2$ there are $\langle \boldsymbol {A}_{i}, F_{i}\rangle , \langle \boldsymbol {A}_{i}, G_{i}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{i})$ such that $\emptyset \ne G_{i} \subsetneq F_{i}$ . To prove this, consider $i = 1, 2$ . Since the logic $\vdash _{i}$ is not truth-equational, we can apply Theorem 5.9 obtaining an algebra $\boldsymbol {A}_{i}$ and distinct $U_{i}, V_{i} \subseteq A_{i}$ such that $\langle \boldsymbol {A}_{i}, U_{i}\rangle , \langle \boldsymbol {A}_{i}, V_{i}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{i})$ . We can assume without loss of generality that $V_{i} \smallsetminus U_{i} \ne \emptyset $ . Bearing this in mind, we define $F_{i} := V_{i}$ and $G_{i} := V_{i} \cap U_{i}$ . It is clear that $\langle \boldsymbol {A}_{i}, G_{i}\rangle \in \mathsf {Mod}(\vdash _{i})$ and $G_{i} \subsetneq F_{i}$ . Since $\vdash _{i}$ has theorems, the fact that $\langle \boldsymbol {A}_{i}, G_{i}\rangle \in \mathsf {Mod}(\vdash _{i})$ implies that $G_{i} \ne \emptyset $ . Moreover, since $G_{i} \subseteq F_{i} = V_{i}$ , we have

Recall that is the identity relation, since $\langle \boldsymbol {A}_{i}, V_{i}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{i})$ . Hence is also the identity relation and, therefore, $\langle \boldsymbol {A}_{i}, G_{i}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{i})$ . Finally, by the assumptions, $\langle \boldsymbol {A}_{i}, F_{i}\rangle = \langle \boldsymbol {A}_{i}, V_{i}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{i})$ . This concludes the proof of the claim.

Together with Lemma 5.7, the claim implies that there are

$$ \begin{align*} \langle \boldsymbol{B}_{1}, F \rangle, \langle \boldsymbol{B}_{1}, G \rangle \in \mathsf{Mod}^{\equiv}(\vdash_{1}) \text{ and }\langle \boldsymbol{B}_{2}, F \rangle, \langle \boldsymbol{B}_{2}, G \rangle \in \mathsf{Mod}^{\equiv}(\vdash_{2}) \end{align*} $$

for some sets $\emptyset \ne G \subsetneq F \subsetneq B$ , an $\mathscr {L}_{\vdash _{1}}$ -algebra $\boldsymbol {B}_{1}$ , and an $\mathscr {L}_{\vdash _{2}}$ -algebra $\boldsymbol {B}_{2}$ such that $B_{1} = B_{2} = B$ .

Now, let $\boldsymbol {B}$ be the common expansion of $\boldsymbol {B}_{1}$ and $\boldsymbol {B}_{2}$ with all finitary operations on B. Moreover, let $\vdash $ be the logic, formulated in $\kappa := \max \{ \vert Fm(\vdash _{1}) \vert , \vert Fm(\vdash _{2}) \vert \}$ variables induced by the set of matrices

$$ \begin{align*} \mathsf{K} := \{ \langle \boldsymbol{B}, F \rangle, \langle \boldsymbol{B}, G \rangle\}. \end{align*} $$

As in the proof of Theorem 5.8, we obtain $\mathsf {K} \subseteq \mathsf {Mod}^{\equiv }(\vdash )$ and ${\vdash _{1}}, {\vdash _{2}} \leqslant {\vdash }$ . Finally, applying Theorem 5.9 to the fact that $\mathsf {K} \subseteq \mathsf {Mod}^{\equiv }(\vdash )$ and $F \ne G$ , we conclude that $\vdash $ is not truth-equational.

Proof Consider two logics $\vdash _{1}$ and $\vdash _{2}$ with theorems that are not assertional. As usual, it will be enough to construct a logic $\vdash $ that is not assertional and in which $\vdash _{1}$ and $\vdash _{2}$ are interpretable.

We claim that for every $i = 1, 2$ there are $\langle \boldsymbol {A}_{i}, F_{i}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{i})$ such that $\vert F_{i} \vert \geqslant 2$ and $F_{i} \subsetneq A_{i}$ . To prove this, consider $i = 1, 2$ , and observe that the fact that $\vdash _{i}$ is not assertional implies that there is $\langle \boldsymbol {A}_{i}, F_{i}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{i})$ such that $F_{i}$ is not a singleton. Since $\vdash _{i}$ has theorems and $\langle \boldsymbol {A}_{i}, F_{i}\rangle $ is a model of $\vdash _{i}$ we know that $F_{i} \ne \emptyset $ and, therefore, $\vert F_{i} \vert \geqslant 2$ . This fact, together with Lemma 5.5, ensures that $F_{i} \subsetneq A_{i}$ , establishing the claim.

By the claim we can find a cardinal $\kappa $ large enough to guarantee that

$$ \begin{align*} \vert A_{1}^{\kappa} \vert = \vert F_{1}^{\kappa} \vert = \vert A_{1}^{\kappa} \smallsetminus F_{1}^{\kappa} \vert = \vert A_{2}^{\kappa} \vert = \vert F_{2}^{\kappa} \vert = \vert A_{2}^{\kappa} \smallsetminus F_{2}^{\kappa} \vert. \end{align*} $$

Thus there is a set B, a set $F \subseteq B$ , an $\mathscr {L}_{\vdash _{1}}$ -algebra $\boldsymbol {B}_{1}$ , and an $\mathscr {L}_{\vdash _{2}}$ -algebra $\boldsymbol {B}_{2}$ such that $B_{1} = B_{2} = B$ and

$$ \begin{align*} \langle \boldsymbol{B}_{1}, F\rangle \cong \langle \boldsymbol{A}_{1}^{\kappa}, F_{1}^{\kappa}\rangle \text{ and }\langle \boldsymbol{B}_{2}, F\rangle \cong \langle \boldsymbol{A}_{2}^{\kappa}, F_{2}^{\kappa}\rangle. \end{align*} $$

Since $\mathsf {Mod}^{\equiv }(\vdash _{i})$ is closed under the formation of direct powers for every $i = 1, 2$ [Reference Jansana and Moraschini24, Lemma 2.3], we obtain $\langle \boldsymbol {B}_{1}, F\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{1})$ and $\langle \boldsymbol {B}_{2}, F\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{2})$ .

Now, let $\boldsymbol {B}$ be the common expansion of $\boldsymbol {B}_{1}$ and $\boldsymbol {B}_{2}$ with all finitary operations on B. Moreover, let $\vdash $ be the logic, formulated in $\kappa := \max \{ \vert Fm(\vdash _{1}) \vert , \vert Fm(\vdash _{2}) \vert \}$ variables induced by the matrix $\langle \boldsymbol {B}, F \rangle $ . As in the proof of Theorem 5.8, we obtain $\langle \boldsymbol {B}, F \rangle \in \mathsf {Mod}^{\equiv }(\vdash )$ and ${\vdash _{1}}, {\vdash _{2}} \leqslant {\vdash }$ . Finally, since $\langle \boldsymbol {B}, F \rangle $ and F is not a singleton, we conclude that $\vdash $ is not assertional.

Proof Let $\vdash _{\mathsf {AI}}$ be an almost inconsistent logic, and recall that if $\vdash $ is a logic without theorems, then ${\vdash } \leqslant {\vdash _{\mathsf {AI}}}$ [Reference Jansana and Moraschini24, Theorem 7.3]. Together with the fact that $\vdash _{\mathsf {IA}}$ lacks theorems, this immediately implies that the Leibniz class of logics with theorems is meet-prime.

Proof Clearly $1$ is a theorem of $\vdash _{\mathsf {BA}}^{\ast }$ . Moreover, the fact that the algebraic reducts of the matrices in $\mathsf {Mod}^{\equiv }(\vdash _{\mathsf {BA}}^{\ast })$ belong to $\mathsf {BA}$ is an immediate consequence of [Reference Jansana and Moraschini24, Corollary 2.6].

It only remains to prove that $\vdash _{\mathsf {BA}}^{\ast }$ is not protoalgebraic. Suppose the contrary, with a view to contradiction. Then there is a set $\varDelta (x, y, \vec {z})$ of congruence formulas with parameters for $\vdash _{\mathsf {BA}}^{\ast }$ . We consider the four-element Boolean algebra $\boldsymbol {A}$ with universe $\{ a, b, 0, 1 \}$ , where $0$ and $1$ are respectively the bottom and the top element of the lattice order. Then we set $F := \{ 1, a \}$ and $G := \{ 1, a, b \}$ . From the definition of $\vdash _{\mathsf {BA}}^{\ast }$ it follows that $\langle \boldsymbol {A}, F \rangle , \langle \boldsymbol {A}, G \rangle \in \mathsf {Mod}(\vdash _{\mathsf {BA}}^{\ast })$ . Together with the fact that $\varDelta $ is a set of congruence formulas with parameters for $\vdash _{\mathsf {BA}}^{\ast }$ and that $F \subseteq G$ , this yields that for every $p, q \in A$ ,

$$ \begin{align*} \langle p, q \rangle \in \boldsymbol{\varOmega}^{\boldsymbol{A}}F \Longleftrightarrow & \, \varDelta^{\boldsymbol{A}}(p, q, \vec{c}) \subseteq F \text{, for every }\vec{c}\in A\\ \Longrightarrow & \, \varDelta^{\boldsymbol{A}}(p, q, \vec{c}) \subseteq G \text{, for every }\vec{c}\in A\\ \Longleftrightarrow & \, \langle p, q \rangle \in \boldsymbol{\varOmega}^{\boldsymbol{A}}G. \end{align*} $$

Hence we conclude that $\boldsymbol {\varOmega }^{\boldsymbol {A}}F \subseteq \boldsymbol {\varOmega }^{\boldsymbol {A}}G$ . On the other hand, it is easy to see that $\boldsymbol {\varOmega }^{\boldsymbol {A}}G$ is the identity relation, while $\boldsymbol {\varOmega }^{\boldsymbol {A}}F$ is the congruence with blocks $\{ 1, a \}$ and $\{ 0, b \}$ . But this contradicts the fact that $\boldsymbol {\varOmega }^{\boldsymbol {A}}F \subseteq \boldsymbol {\varOmega }^{\boldsymbol {A}}G$ , as desired.

Proof We set

$$ \begin{align*} \mathbb{K}_{1} := \mathsf{Proto} \lor \mathsf{Asrt} \;\; \text{ and } \;\; \mathbb{K}_{2} := \mathsf{Proto} \lor\mathrm{Log}(\vdash_{\mathsf{BA}}^{\ast}). \end{align*} $$

Observe that $\mathbb {K}_{1}$ and $\mathbb {K}_{2}$ are Leibniz classes by Theorems 3.4 and 3.11. Moreover, they comprise only logics with theorems by Propositions 2.6 and 3.15. We have

(15) $$ \begin{align} \mathsf{Proto} \subsetneq \mathbb{K}_{1} \;\; \text{ and }\;\; \mathsf{Equiv} \subsetneq \mathbb{K}_{2}. \end{align} $$

The validity of the inclusion $\mathsf {Proto} \subseteq \mathbb {K}_{1} \cap \mathbb {K}_{2}$ is straightforward. The fact that $\mathsf {Proto} \subsetneq \mathbb {K}_{1}$ is witnessed by the existence of assertional logics that are not protoalgebraic [Reference Raftery32, Example 7]. Finally, the fact that the inclusion $\mathsf {Equiv} \subseteq \mathbb {K}_{2}$ is strict follows from Lemma 6.2.

In the light of (15), it only remains to prove that $\mathbb {K}_{1} \cap \mathbb {K}_{2} \subseteq \mathsf {Proto}$ . Suppose the contrary, with a view to contradiction. Then there is a logic ${\vdash } \in \mathbb {K}_{1} \cap \mathbb {K}_{2}$ that is not protoalgebraic. By Proposition 2.6, and considering that the non-indexed product of two protoalgebraic logics is protoalgebraic and interpretable in each of them, there are a protoalgebraic logic $\vdash _{pr}$ and an assertional logic $\vdash _{as}$ such that

(16) $$ \begin{align} {\vdash_{pr} \bigotimes \vdash_{as}} \leqslant {\vdash} \;\; \text{ and }\:\; {\vdash_{pr} \bigotimes \vdash_{\mathsf{BA}}^{\ast}} \leqslant {\vdash}. \end{align} $$

Since $\vdash $ has theorems and is not protoalgebraic, we can apply Theorem 3.1(ii) obtaining that $\mathsf {Mod}^{\equiv }(\vdash ) \ne \mathbb {R}(\mathsf {Mod}^{\equiv }(\vdash ))$ , i.e., that there is a matrix $\langle \boldsymbol {A}, F\rangle \in \mathsf {Mod}^{\equiv }(\vdash )$ such that $\boldsymbol {\varOmega }^{\boldsymbol {A}}F$ is not the identity relation.

We claim that for every pair of different $a, c \in A$ , if $\langle a, c \rangle \in \boldsymbol {\varOmega }^{\boldsymbol {A}}F$ , then $a, c \notin F$ . To prove this, consider different $a, c \in A$ such that $\langle a, c \rangle \in \boldsymbol {\varOmega }^{\boldsymbol {A}}F$ . By (16) there is an interpretation $\boldsymbol {\tau }$ of $\vdash _{pr} \bigotimes \vdash _{as}$ into $\vdash $ . By Proposition 6.1 we obtain without loss of generality thatFootnote ⁴

$$ \begin{align*} \langle \boldsymbol{A}^{\boldsymbol{\tau}}, F \rangle = \langle \boldsymbol{A}_{1} \bigotimes \boldsymbol{A}_{2}, F_{1} \times F_{2}\rangle \end{align*} $$

for some $\langle \boldsymbol {A}_{1}, F_{1}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{pr})$ and $\langle \boldsymbol {A}_{2}, F_{2}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{as})$ . Since $a, c \in A = A_{1} \times A_{2}$ , there are $a_{1}, c_{1} \in A_{1}$ and $a_{2}, c_{2} \in A_{2}$ such that

$$ \begin{align*} a = \langle a_{1}, a_{2}\rangle \text{ and }c = \langle c_{1}, c_{2}\rangle. \end{align*} $$

From Proposition 6.1(ii) it follows that there are $\theta _{1} \in \text {Con} \boldsymbol {A}_{1}$ and $\theta _{2} \in \text {Con} \boldsymbol {A}_{2}$ compatible with $F_{1}$ and $F_{2}$ , respectively, such that

(17) $$ \begin{align} \boldsymbol{\varOmega}^{\boldsymbol{A}}F = \{ \langle \langle p, q \rangle, \langle r, s \rangle \rangle \colon \langle p, r \rangle \in \theta_{1} \text{ and }\langle q, s \rangle \in \theta_{2}\}. \end{align} $$

Since $\vdash _{pr}$ is protoalgebraic and $\langle \boldsymbol {A}_{1}, F_{1} \rangle \in \mathsf {Mod}^{\equiv }(\vdash _{pr})$ , we can apply Theorem 3.1(ii) obtaining that $\boldsymbol {\varOmega }^{\boldsymbol {A}_{1}}F_{1}$ is the identity relation. Together with the fact that $\theta _{1} \subseteq \boldsymbol {\varOmega }^{\boldsymbol {A}_{1}}F_{1}$ and (17), this yields

(18) $$ \begin{align} \boldsymbol{\varOmega}^{\boldsymbol{A}}F = \{ \langle \langle p, q \rangle, \langle p, s \rangle \rangle \colon p \in A_{1} \text{ and }\langle q, s \rangle \in \theta_{2}\}. \end{align} $$

Since $\langle a, c \rangle \in \boldsymbol {\varOmega }^{\boldsymbol {A}}F$ and $a \ne c$ , we conclude that $a_{2} \ne c_{2}$ and $\langle a_{2}, c_{2} \rangle \in \theta _{2}$ .

Now, from the fact that $\theta _{2}$ is compatible with $F_{2}$ and $\langle a_{2}, c_{2}\rangle \in \theta _{2}$ , it follows

(19) $$ \begin{align} \text{either }a_{2}, c_{2} \in F_{2} \text{ or }a_{2}, c_{2} \notin F_{2}. \end{align} $$

Since $\vdash _{as}$ is assertional and $\langle \boldsymbol {A}_{2}, F_{2}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{as})$ , we know that $F_{2}$ is a singleton. Together with $a_{2} \ne c_{2}$ and (19), this yields that $a_{2}, c_{2} \notin F_{2}$ . As a consequence, we obtain $\langle a, c \rangle \notin F_{1} \times F_{2} = F$ , establishing the claim.

Now, by (16) there is an interpretation $\boldsymbol {\tau }$ of $\vdash _{pr} \bigotimes \vdash _{\mathsf {BA}}^{\ast }$ into $\vdash $ . By Proposition 6.1 we obtain without loss of generality that

$$ \begin{align*} \langle \boldsymbol{A}^{\boldsymbol{\tau}}, F \rangle = \langle \boldsymbol{A}_{1} \bigotimes \boldsymbol{A}_{2}, F_{1} \times F_{2}\rangle \end{align*} $$

for some $\langle \boldsymbol {A}_{1}, F_{1}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{pr})$ and $\langle \boldsymbol {A}_{2}, F_{2}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{\mathsf {BA}}^{\ast })$ .

Recall that the matrix $\langle \boldsymbol {A}, F \rangle $ is not reduced. Then there are different $a, c \in A$ such that $\langle a, c\rangle \in \boldsymbol {\varOmega }^{\boldsymbol {A}}F$ . Since $A = A_{1} \times A_{2}$ , there are $a_{1}, c_{1} \in A_{1}$ and $a_{2}, c_{2} \in A_{2}$ such that

$$ \begin{align*} a = \langle a_{1}, a_{2}\rangle \text{ and }c = \langle c_{1}, c_{2}\rangle. \end{align*} $$

As in the proof of the claim, we obtain

(20) $$ \begin{align} \boldsymbol{\varOmega}^{\boldsymbol{A}}F = \{ \langle \langle p, q \rangle, \langle p, s \rangle \rangle \colon p \in A_{1} \text{ and }\langle q, s \rangle \in \theta\} \end{align} $$

for some $\theta \in \text {Con}\boldsymbol {A}_{2}$ compatible with $F_{2}$ . Then we choose elements $e_{1} \in F_{1}$ and $e_{2} \in F_{2}$ (the fact that the logics $\vdash _{pr}$ and $\vdash _{\mathsf {BA}}^{\ast }$ have theorems guarantees $F_{1}, F_{2} \ne \emptyset $ ). Clearly we have

(21) $$ \begin{align} \langle e_{1}, e_{2}\rangle \in F_{1} \times F_{2} = F. \end{align} $$

Together with $a \ne c$ and $\langle a, c \rangle \in \boldsymbol {\varOmega }^{\boldsymbol {A}}F$ , the display (20) implies $a_{2} \ne c_{2}$ and $\langle a_{2}, c_{2}\rangle \in \theta $ . Since $\langle \boldsymbol {A}_{2}, F_{2}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{\mathsf {BA}}^{\ast })$ , we can apply Lemma 6.2 obtaining that $\boldsymbol {A}_{2}$ is a Boolean algebra. Since $\boldsymbol {A}_{2}$ is congruence uniform and $\theta $ identifies two distinct elements of $A_{2}$ (namely $a_{2}$ and $c_{2}$ ), then there is $b \in A_{2} \smallsetminus \{ e_{2} \}$ such that $\langle e_{2}, b \rangle \in \theta $ . By (20) we conclude

$$ \begin{align*} \langle \langle e_{1}, e_{2}\rangle, \langle e_{1}, b \rangle \rangle \in \boldsymbol{\varOmega}^{\boldsymbol{A}}F \text{ and } \langle e_{1}, e_{2}\rangle \ne \langle e_{1}, b \rangle. \end{align*} $$

Together with (21) this contradicts the claim. This produces the desired contradiction.

Fact 1. Logics with an injective theorem form a Leibniz class.

Proof It is not hard to show that logics with an injective theorem are closed under term-equivalence, compatible expansions, and non-indexed products of sets. In the light of Theorem 2.2 this implies that they form a Leibniz class.

Fact 2. The logic $\vdash _{\nabla }$ is protoalgebraic with set of congruence formulas with parameters

$$ \begin{align*} \hat{\nabla}(x, y, \vec{z}) := \{ \varphi(x, \vec{z}) \to \varphi(y, \vec{z}) \colon \varphi(x, \vec{z}) \in Fm(\vdash) \}. \end{align*} $$

Proof The logic $\vdash _{\nabla }$ is protoalgebraic, since the set $\nabla (x, y)$ satisfies the requirements of Theorem 3.1(iv). By the same theorem, $\hat {\nabla }$ is a set of congruence formulas for $\vdash _{\nabla }$ .

Fact 3. For every formula $\varphi \in Fm(\vdash _{\nabla })$ ,

$$ \begin{align*} \emptyset \vdash_{\nabla}\varphi \Longleftrightarrow \varphi = \psi \to \psi\text{, for some formula }\psi. \end{align*} $$

Proof See [Reference Font16, Proposition 2.1].

Fact 4. The logic $\vdash _{\varDelta }$ is protoalgebraic.

Proof This is a consequence of Fact 2, together with the fact that protoalgebraicity is preserved by extensions.

Fact 5. The formula $x \to x$ is an injective theorem of $\vdash _{\varDelta }$ .

Proof Clearly $x \to x$ is a theorem of $\vdash _{\varDelta }$ . Then consider $\langle \boldsymbol {A}, F \rangle \in \mathsf {Mod}^{\equiv }(\vdash _{\varDelta })$ and $a, b \in A$ such that $a \to ^{\boldsymbol {A}} a = b\to ^{\boldsymbol {A}}b$ . We need to show that $a = b$ . To this end, recall by Fact 4 that $\vdash _{\varDelta }$ is protoalgebraic. Then we can apply Theorem 3.1(iii), obtaining that the matrix $\langle \boldsymbol {A}, F \rangle $ is reduced, i.e., that the congruence $\boldsymbol {\varOmega }^{\boldsymbol {A}}F$ is the identity relation. Therefore, to conclude the proof, it will be enough to show that $\langle a, b \rangle \in \boldsymbol {\varOmega }^{\boldsymbol {A}}F$ . By [Reference Jansana and Moraschini24, Proposition 2.2(i)] this amounts to establishing that for every formula $\varphi (x, \vec {z}) \in Fm(\vdash _{\varDelta })$ and every $\vec {c} \in A$ ,

(23) $$ \begin{align} \varphi^{\boldsymbol{A}}(a, \vec{c}) \in F \Longleftrightarrow \varphi^{\boldsymbol{A}}(b, \vec{c}) \in F. \end{align} $$

To prove the implication from left to right, consider $\varphi (x, \vec {z}) \in Fm(\vdash _{\varDelta })$ and $\vec {c} \in A$ such that $\varphi ^{\boldsymbol {A}}(a, \vec {c}) \in F$ . Consider also an arbitrary formula $\psi (x, \vec {z}) \in Fm(\vdash _{\nabla })$ . We have that $\psi (x, \vec {z}) \to \psi (x, \vec {z})$ is a theorem of $\vdash _{\varDelta }$ . Together with the fact that $a \to ^{\boldsymbol {A}} a = b\to ^{\boldsymbol {A}}b$ , this yields

$$ \begin{align*} \psi^{\boldsymbol{A}}(a \to a, \vec{c}) \to \psi^{\boldsymbol{A}}(b \to b, \vec{c}) = \psi^{\boldsymbol{A}}(a \to a, \vec{c}) \to \psi^{\boldsymbol{A}}(a \to a, \vec{c}) \in F. \end{align*} $$

Hence we conclude that $\hat {\nabla }(a \to ^{\boldsymbol {A}} a, b \to ^{\boldsymbol {A}}b, \vec {c}) \subseteq F$ . Since $\langle \boldsymbol {A}, F \rangle $ is a model of (22), this implies $\hat {\nabla }(a, b, \vec {c}) \subseteq F$ . As a consequence, we obtain $\varphi ^{\boldsymbol {A}}(a, \vec {c}) \to \varphi ^{\boldsymbol {A}}(b, \vec {c}) \in F$ . Together with the assumption that $\varphi ^{\boldsymbol {A}}(a, \vec {c}) \in F$ , and the fact that $\langle \boldsymbol {A}, F \rangle $ is a model of the rule $x, x \to y \rhd y$ , this implies $\varphi ^{\boldsymbol {A}}(b, \vec {c}) \in F$ . This concludes the proof of the left to right implication in (23). The proof of the other implication is analogous. From (23) it follows that $\langle a, b \rangle \in \boldsymbol {\varOmega }^{\boldsymbol {A}}F$ and, therefore, $a = b$ .

Fact 6. The rules in (22) are admissible in $\vdash _{\nabla }$ .

Proof To prove this, consider a substitution $\sigma $ such that $\emptyset \vdash _{\nabla } \sigma [\hat {\nabla }( x \to x, y \to y, \vec {z})]$ . Since $x \to y \in \hat {\nabla }(x, y, \vec {z})$ , we have $(x \to x) \to (y \to y) \in \hat {\nabla }(x \to x, y \to y, \vec {z})$ . As a consequence, we obtain $\emptyset \vdash _{\nabla }(\sigma x \to \sigma x) \to (\sigma y \to \sigma y)$ . By Fact 3 this implies $\sigma x \to \sigma x = \sigma y \to \sigma y$ and, therefore, $\sigma x = \sigma y$ . Since $\emptyset \vdash _{\nabla } x \to x$ and $\sigma x = \sigma y$ , we obtain $\emptyset \vdash _{\nabla } \sigma \varphi (x, \vec {z}) \to \sigma \varphi (y, \vec {z})$ , for every formula $\varphi (x, \vec {z}) \in Fm(\vdash _{\nabla })$ . But this amounts to the fact that $\emptyset \vdash _{\nabla } \sigma [\hat {\nabla }( x, y, \vec {z})]$ . Hence we conclude that the rules in (22) are admissible in $\vdash _{\nabla }$ .

Fact 7. The logic $\vdash _{\varDelta }$ is not equivalential.

Proof Suppose, with a view to contradiction, that $\vdash _{\varDelta }$ is equivalential. Then $\vdash _{\varDelta }$ has a set of congruence formulas $\varDelta (x, y)$ .

We claim that $\emptyset \vdash _{\varDelta }\varphi $ , for every formula $\varphi $ such that $\varDelta (x, y) \vdash _{\varDelta } \varphi $ and ${Var}(\varphi ) \nsubseteq \{ x, y \}$ . To demonstrate this, we reason by complete induction on the length of proofs in $\vdash _{\varDelta }$ . Consider an ordinal $\alpha $ , and suppose that $\emptyset \vdash _{\varDelta } \psi $ , for every formula $\psi $ such that ${Var}(\varphi ) \nsubseteq \{ x, y \}$ , and of which the exists a proof indexed by an ordinal $< \alpha $ from $\varDelta (x, y)$ . Now, let $\pi := \{ \gamma _{\beta } \colon \beta < \alpha \}$ be a proof of a formula $\varphi $ such that ${Var}(\varphi ) \nsubseteq \{ x, y \}$ from $\varDelta (x, y)$ . If $\varphi $ is a substitution instance of the axiom $x \to x$ , then it is a theorem and we are done. Moreover, observe that $\varphi \notin \varDelta (x, y)$ , since ${Var}(\varphi ) \nsubseteq \{ x, y \}$ . Therefore, $\varphi $ is obtained by the application of one of the inference rules of $\vdash _{\varDelta }$ to a proper initial segment of $\pi $ .

First consider the case where $\varphi $ is obtained by an application of the rule $x, x\to y \rhd y$ . Then there is some formula $\psi $ such that $\psi $ and $\psi \to \varphi $ appear in a proper initial segment of $\pi $ . Since ${Var}(\varphi ) \nsubseteq \{ x, y \}$ , then ${Var}(\psi \to \varphi ) \subseteq \{ x, y \}$ . Therefore we can apply the induction hypothesis obtaining $\emptyset \vdash _{\varDelta } \psi \to \varphi $ . By Fact 6 we get $\emptyset \vdash _{\nabla } \psi \to \varphi $ . Moreover, by Fact 3 this yields $\psi = \varphi $ . In particular, this implies ${Var}(\psi ) \nsubseteq \{ x, y \}$ . Therefore we can apply the induction hypothesis, obtaining $\emptyset \vdash _{\varDelta }\psi $ . Since $\varphi = \psi $ , we conclude that $\emptyset \vdash _{\varDelta }\varphi $ , as desired.

Then we consider the case where $\varphi $ is obtained by an application of one of the rules in (22). Then there is a substitution $\sigma $ and a formula $\psi (x, y, \vec {z}) \in \hat {\nabla }( x, y, \vec {z})$ such that $\varphi = \sigma \psi $ , and each element of $\sigma [\hat {\nabla }( x \to x, y \to y, \vec {z})]$ appears in a proper initial segment of $\pi $ . Suppose, with a view to contradiction, that $\emptyset \nvdash _{\varDelta } \sigma \psi $ . Since $\psi \in \hat {\nabla }(x, y, \vec {z})$ , there is a formula $\gamma (x, y, \vec {z}) \in Fm(\vdash _{\nabla })$ such that $\psi = \gamma (x, y, \vec {z}) \to \gamma (y, y, \vec {z})$ . Together with the fact that $\emptyset \vdash _{\varDelta } x \to x$ and $\emptyset \nvdash _{\varDelta } \sigma \psi $ , this yields $\sigma (\gamma (x, y, \vec {z})) \ne \sigma (\gamma (y, y, \vec {z}))$ . But this easily implies $\sigma x \ne \sigma y$ and $x \in {Var}(\gamma (x, y, \vec {z}))$ . As a consequence we obtain $\sigma (\gamma (x \to x, y \to y, \vec {z})) \ne \sigma (\gamma (y \to y, y \to y, \vec {z}))$ . Moreover, from Facts 6 and 3, this yields

$$ \begin{align*} \emptyset \nvdash_{\varDelta}\sigma(\gamma(x \to x, y \to y, \vec{z})) \to \sigma(\gamma(y \to y, y \to y, \vec{z})), \end{align*} $$

which amounts to

(24) $$ \begin{align} \emptyset \nvdash_{\varDelta} \sigma \psi(x\to x, y, \to y, \vec{z})\text{, where }\psi = \psi(x, y, \vec{z}). \end{align} $$

Now, from the fact that ${Var}(\sigma \psi ) = {Var}(\varphi ) \nsubseteq \{ x, y \}$ it follows

$$ \begin{align*} {Var}(\sigma \psi(x \to x, y \to y, \vec{z})) \nsubseteq \{ x, y \}. \end{align*} $$

The above display and the fact that $\sigma \psi (x \to x, y \to y, \vec {z}) \in \sigma [\hat {\nabla }(x \to x, y \to y, \vec {z})]$ allow us to apply the induction hypothesis, obtaining

$$ \begin{align*} \emptyset \vdash_{\varDelta}\sigma \psi(x \to x, y \to y, \vec{z}). \end{align*} $$

But this contradicts (24). Hence we conclude that $\emptyset \vdash _{\varDelta } \sigma \psi $ , establishing the claim.

Now we move back to the main proof. First observe that

(25) $$ \begin{align} \varDelta(x, y) \vdash_{\varDelta} (x \to z) \to (y \to z). \end{align} $$

To prove this, consider $\langle \boldsymbol {A}, F \rangle \in \mathsf {Mod}(\vdash _{\varDelta })$ and $a, b, c \in A$ such that $\varDelta ^{\boldsymbol {A}}(a, b) \subseteq F$ . Since $\varDelta $ is a set of congruence formulas for $\vdash _{\varDelta }$ , this yields $\langle a, b \rangle \in \boldsymbol {\varOmega }^{\boldsymbol {A}}F$ and, therefore,

$$ \begin{align*} \langle (a \to c) \to (a \to c), (a \to c) \to (b \to c) \rangle \in \boldsymbol{\varOmega}^{\boldsymbol{A}}F. \end{align*} $$

Since $x \to x$ is a theorem of $\vdash _{\varDelta }$ , we have that $(a \to c) \to (a \to c) \in F$ . As $\boldsymbol {\varOmega }^{\boldsymbol {A}}F$ is compatible with F, this implies $(a \to c) \to (b \to c) \in F$ , establishing (25). From (25) and the claim it follows that $\emptyset \vdash _{\varDelta } (x \to z) \to (y \to z)$ . By Facts 6 and 3 this yields $x \to z = y \to z$ , which is false.

Proof We begin by setting

$$ \begin{align*} \mathbb{K}_{1} := \mathsf{Equiv} \lor (\mathsf{Proto} \cap \mathsf{Injctv}) \; \text{ and }\; \mathbb{K}_{2} := \mathsf{Equiv} \lor (\mathsf{Proto} \cap \mathsf{Asrt}). \end{align*} $$

The fact that $\mathbb {K}_{1}$ and $\mathbb {K}_{2}$ are Leibniz classes is a consequence of Theorems 3.4, 3.7, and 3.11, and Proposition 6.5. Moreover, they comprise only logics with theorems by Propositions 2.6 and 3.15. We have

(26) $$ \begin{align} \mathsf{Equiv} \subsetneq \mathbb{K}_{1} \; \text{ and } \; \mathsf{Equiv} \subsetneq \mathbb{K}_{2}. \end{align} $$

The validity of the inclusion $\mathsf {Equiv} \subseteq \mathbb {K}_{1} \cap \mathbb {K}_{2}$ is straightforward. Moreover, Proposition 6.5 guarantees that $\mathsf {Equiv} \subsetneq \mathbb {K}_{1}$ . Finally, the fact that the inclusion $\mathsf {Equiv} \subseteq \mathbb {K}_{2}$ is strict follows from the following observation [Reference Czelakowski and Jansana13, Proposition 6.1 and Theorem 6.4]:

$$ \begin{align*} (\mathsf{Proto} \cap \mathsf{Asrt}) \smallsetminus \mathsf{Equiv} \ne \emptyset. \end{align*} $$

In the light of (26), it only remains to prove $\mathbb {K}_{1} \cap \mathbb {K}_{2} \subseteq \mathsf {Equiv}$ . Suppose the contrary, with a view to contradiction. Then there is a logic ${\vdash } \in \mathbb {K}_{1} \cap \mathbb {K}_{2}$ that is not equivalential. By Proposition 2.6 there are an equivalential logic $\vdash _{eq}$ , a protoalgebraic logic with an injective theorem $\vdash _{pin}$ , and an assertional protoalgebraic logic $\vdash _{ap}$ such that

(27) $$ \begin{align} {\vdash_{eq} \bigotimes \vdash_{pin}} \leqslant {\vdash} \; \text{ and } \; {\vdash_{eq} \bigotimes \vdash_{ap}} \leqslant {\vdash}. \end{align} $$

Together with [Reference Jansana and Moraschini24, Proposition 3.8] this implies that $\vdash $ is term-equivalent to a compatible expansion of the non-indexed product of a pair of protoalgebraic logics. Since protoalgebraic logics form a Leibniz class, by Theorem 2.2(ii) we conclude that $\vdash $ is protoalgebraic. Moreover, since $\vdash $ is not equivalential, we can apply Theorem 6.6 obtaining that $\mathsf {Mod}^{\equiv }(\vdash )$ is not closed under $\mathbb {S}$ , i.e., there is a matrix $\langle \boldsymbol {A}, F\rangle \in \mathsf {Mod}^{\equiv }(\vdash )$ with a submatrix $\langle \boldsymbol {B}, F \cap B \rangle \subseteq \langle \boldsymbol {A}, F\rangle $ such that $\langle \boldsymbol {B}, F \cap B \rangle \notin \mathsf {Mod}^{\equiv }(\vdash )$ .

We claim that for every pair of different $a, c \in B$ , if $\langle a, c \rangle \in \boldsymbol {\varOmega }^{\boldsymbol {B}}(F \cap B)$ , then $a, b \notin F$ . To prove this, consider different $a, c \in B$ such that $\langle a, c \rangle \in \boldsymbol {\varOmega }^{\boldsymbol {B}}(F \cap B)$ . By (28) there is an interpretation $\boldsymbol {\tau }$ of $\vdash _{eq} \bigotimes \vdash _{ap}$ into $\vdash $ . By Proposition 6.1(i) we obtain without loss of generality that

$$ \begin{align*} \langle \boldsymbol{A}^{\boldsymbol{\tau}}, F \rangle &= \langle \boldsymbol{A}_{1} \bigotimes \boldsymbol{A}_{2}, F_{1} \times F_{2}\rangle,\\ \langle \boldsymbol{B}^{\boldsymbol{\tau}}, F \cap B \rangle &= \langle \boldsymbol{B}_{1} \bigotimes \boldsymbol{B}_{2}, (F_{1} \cap B_{1}) \times (F_{2} \cap B_{2}) \rangle \end{align*} $$

for some $\langle \boldsymbol {B}_{1}, F_{1}\rangle \subseteq \langle \boldsymbol {A}_{1}, F_{1}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{eq})$ and $\langle \boldsymbol {B}_{2}, F_{2}\rangle \subseteq \langle \boldsymbol {A}_{2}, F_{2}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{pa})$ . Since $a, c \in B = B_{1} \times B_{2}$ , there are $a_{1}, c_{1} \in B_{1}$ and $a_{2}, c_{2} \in B_{2}$ such that

$$ \begin{align*} a = \langle a_{1}, a_{2}\rangle \; \text{ and } \; c = \langle c_{1}, c_{2}\rangle. \end{align*} $$

Again from Proposition 6.1(ii) it follows that there are $\theta _{1} \in \text {Con} \boldsymbol {B}_{1}$ and $\theta _{2} \in \text {Con} \boldsymbol {B}_{2}$ compatible with $F_{1} \cap B_{1}$ and $F_{2} \cap B_{2}$ , respectively, such that

$$ \begin{align*} \boldsymbol{\varOmega}^{\boldsymbol{B}}(F \cap B) = \{ \langle \langle p, q \rangle, \langle r, s \rangle \rangle \colon \langle p, r \rangle \in \theta_{1} \text{ and }\langle q, s \rangle \in \theta_{2}\}. \end{align*} $$

Together with the fact that $\langle a, c \rangle \in \boldsymbol {\varOmega }^{\boldsymbol {B}}(F \cap B)$ , this yields

(28) $$ \begin{align} \langle a_{i}, c_{i}\rangle \in \theta_{i} \subseteq \boldsymbol{\varOmega}^{\boldsymbol{B}_{i}}(F_{i} \cap B_{i}), \text{ for every }i = 1, 2. \end{align} $$

Since $\vdash _{eq}$ is equivalential, we can apply Theorem 6.6 obtaining that $\mathsf {Mod}^{\equiv }(\vdash _{eq})$ is closed under $\mathbb {S}$ . Together with $\langle \boldsymbol {A}_{1}, F_{1}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{eq})$ , this implies $\langle \boldsymbol {B}_{1}, F_{1} \cap B_{1}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{eq})$ . Now, recall from Theorem 3.1(ii) that $\mathsf {Mod}^{\equiv }(\vdash _{eq}) = \mathbb {R}(\mathsf {Mod}(\vdash _{eq}))$ . As a consequence, the congruence $\boldsymbol {\varOmega }^{\boldsymbol {B}_{1}}(F_{1} \cap B_{1})$ is the identity relation. Hence by (28) we get $a_{1} = c_{1}$ . Since $a \ne c$ , we conclude that $a_{2} \ne c_{2}$ .

Now, from (28) and the fact that $\boldsymbol {\varOmega }^{\boldsymbol {B}_{2}}(F_{2} \cap B_{2})$ is compatible with $F_{2} \cap B_{2}$ it follows that

(29) $$ \begin{align} \text{either }a_{2}, c_{2} \in F_{2} \cap B_{2} \text{ or }a_{2}, c_{2} \notin F_{2} \cap B_{2}. \end{align} $$

Since $\vdash _{pa}$ is assertional and $\langle \boldsymbol {A}_{2}, F_{2}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{pa})$ , we know that $F_{2}$ is a singleton. Together with $a_{2} \ne c_{2}$ and (29), this yields $a_{2}, c_{2} \notin F_{2} \cap B_{2}$ . As a consequence, we obtain $\langle a, c \rangle \notin F_{1} \times F_{2} = F$ , establishing the claim.

Now, by (28) there is an interpretation $\boldsymbol {\tau }$ of $\vdash _{eq} \bigotimes \vdash _{pin}$ into $\vdash $ . By Proposition 6.1(i) we obtain without loss of generality that

$$ \begin{align*} \langle \boldsymbol{A}^{\boldsymbol{\tau}}, F \rangle &= \langle \boldsymbol{A}_{1} \bigotimes \boldsymbol{A}_{2}, F_{1} \times F_{2}\rangle,\\ \langle \boldsymbol{B}^{\boldsymbol{\tau}}, F \cap B \rangle &= \langle \boldsymbol{B}_{1} \bigotimes \boldsymbol{B}_{2}, (F_{1} \cap B_{1}) \times (F_{2} \cap B_{2}) \rangle \end{align*} $$

for some $\langle \boldsymbol {B}_{1}, F_{1}\rangle \subseteq \langle \boldsymbol {A}_{1}, F_{1}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{eq})$ and $\langle \boldsymbol {B}_{2}, F_{2}\rangle \subseteq \langle \boldsymbol {A}_{2}, F_{2}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{pin})$ .

Recall that the matrix $\langle \boldsymbol {B}, F \cap B \rangle $ is not reduced. Then there are different $a, c \in B$ such that $\langle a, c\rangle \in \boldsymbol {\varOmega }^{\boldsymbol {B}}(F \cap B)$ . Since $B = B_{1} \times B_{2}$ , there are $a_{1}, c_{1} \in B_{1}$ and $a_{2}, c_{2} \in B_{2}$ such that

$$ \begin{align*} a = \langle a_{1}, a_{2}\rangle \; \text{ and } \; c = \langle c_{1}, c_{2}\rangle. \end{align*} $$

As in the proof of the claim, we obtain $a_{2} \ne c_{2}$ and $\langle a_{2}, c_{2}\rangle \in \boldsymbol {\varOmega }^{\boldsymbol {B}_{2}}(F_{2} \cap B_{2})$ .

Now, let $\varphi (x)$ be an injective theorem of $\vdash _{pin}$ . Since $\langle \boldsymbol {A}_{2}, F_{2}\rangle \in \mathsf {Mod}^{\equiv }(\vdash _{pin})$ and $a_{2} \ne c_{2}$ , we have

(30) $$ \begin{align} \varphi^{\boldsymbol{A}_{2}}(a_{2}) \ne \varphi^{\boldsymbol{A}_{1}}(a_{2}) \text{ and }\varphi^{\boldsymbol{A}_{2}}(a_{2}), \varphi^{\boldsymbol{A}_{2}}(c_{2}) \in F_{2}. \end{align} $$

Let also $\top (x)$ be an arbitrary theorem of $\vdash _{eq}$ . Since $\langle \boldsymbol {A}_{1}, F_{1}\rangle \in \mathsf {Mod}(\vdash _{eq})$ , we have

(31) $$ \begin{align} \top^{\boldsymbol{A}_{1}}(a_{1}), \top^{\boldsymbol{A}_{1}}(c_{1}) \in F_{1}. \end{align} $$

Observe that the pair $\langle \top , \varphi \rangle $ is a unary connective of ${\vdash _{eq}} \bigotimes {\vdash _{pld}}$ . Together with (30, 31), this yields

$$ \begin{align*} \langle \top, \varphi \rangle^{\boldsymbol{B}_{1} \bigotimes \boldsymbol{B}_{2}} (a)&= \langle \top^{\boldsymbol{A}_{1}}(a_{1}), \varphi^{\boldsymbol{A}_{2}}(a_{2})\rangle \in F_{1} \times F_{2} = F,\\ \langle \top, \varphi \rangle^{\boldsymbol{B}_{1} \bigotimes \boldsymbol{B}_{2}} (c)&= \langle \top^{\boldsymbol{A}_{1}}(c_{1}), \varphi^{\boldsymbol{A}_{2}}(c_{2})\rangle \in F_{1} \times F_{2} = F \end{align*} $$

and

$$ \begin{align*} \langle \top, \varphi \rangle^{\boldsymbol{B}_{1} \bigotimes \boldsymbol{B}_{2}} (a) \ne \langle \top, \varphi \rangle^{\boldsymbol{B}_{1} \bigotimes \boldsymbol{B}_{2}} (c). \end{align*} $$

Since $\boldsymbol {B}_{1} \bigotimes \boldsymbol {B}_{2} = \boldsymbol {B}^{\boldsymbol {\tau }}$ , we know that $\langle \top , \varphi \rangle ^{\boldsymbol {B}_{1} \bigotimes \boldsymbol {B}_{2}}$ is a term-function of $\boldsymbol {B}$ . Together with the above displays and the fact that $\langle a, c \rangle \in \boldsymbol {\varOmega }^{\boldsymbol {B}}(F \cap B)$ , this implies that $\boldsymbol {\varOmega }^{\boldsymbol {B}}(F \cap B)$ identifies two different elements of F, i.e., $\langle \top , \varphi \rangle ^{\boldsymbol {B}_{1} \bigotimes \boldsymbol {B}_{2}} (a)$ and $\langle \top , \varphi \rangle ^{\boldsymbol {B}_{1} \bigotimes \boldsymbol {B}_{2}} (c)$ . But this contradicts the claim. Hence we reached a contradiction, as desired.

Proof First we set $\mathbb {K} := \mathsf {Order} \lor \mathsf {Truth}$ . Observe that $\mathbb {K}$ and $\mathsf {Equiv}$ are Leibniz classes by Theorems 3.7, 3.12, and 3.13. The fact that they comprise only logics with theorems is a consequence of Propositions 2.6 and 3.15. Moreover, we have

(32) $$ \begin{align} \mathsf{Order} \subsetneq \mathbb{K}\;\; \text{ and }\;\; \mathsf{Order} \subsetneq \mathsf{Equiv}. \end{align} $$

To prove this, recall that every order algebraizable logic is equivalential [Reference Raftery33, Proposition 7.11(iii)]. In particular, this implies that $\mathsf {Order} \subseteq \mathbb {K} \cap \mathsf {Equiv}$ . The fact that the inclusion $\mathsf {Order} \subseteq \mathbb {K}$ is strict is an immediate consequence of the fact that so is the inclusion $\mathsf {Equiv} \subseteq \mathbb {K}_{2}$ in the proof of Theorem 6.7. On the other hand, the fact that $\mathsf {Order} \subsetneq \mathsf {Equiv}$ is witnessed by the existence of equivalential logics that are not order algebraizable [Reference Raftery33, p. 267].

In the light of (32), it only remains to prove that $\mathbb {K} \cap \mathsf {Equiv} \subseteq \mathsf {Order}$ . To this end, consider a logic ${\vdash } \in \mathbb {K} \cap \mathsf {Equiv}$ . Clearly, $\vdash $ is equivalential and, therefore, there is a set of formulas $\varDelta (x, y)$ of $\vdash $ such that for every $\langle \boldsymbol {A}, F \rangle \in \mathsf {Mod}^{\equiv }(\vdash )$ and $a, c \in A$ ,

(33) $$ \begin{align} a = c \Longleftrightarrow \varDelta^{\boldsymbol{A}}(a, c) \subseteq F. \end{align} $$

Moreover, since ${\vdash } \in \mathbb {K}$ , there are an order algebraizable logic $\vdash _{or}$ , a truth-equational logic $\vdash _{tr}$ , and an interpretation $\boldsymbol {\tau }$ of $\vdash _{or} \bigotimes \vdash _{tr}$ into $\vdash $ . Then there there are a set of formulas $\nabla (x, y)$ and a set of inequalities $I(x)$ of $\vdash _{or}$ such that for every $\langle \boldsymbol {A}, F \rangle \in \mathsf {Mod}^{\equiv }(\vdash _{or})$ ,

1. (i) the relation $\preccurlyeq ^{\boldsymbol {A}}_{F}$ on A defined for every $a, c \in A$ by

   $$ \begin{align*} a \preccurlyeq_{F}^{\boldsymbol{A}} c \Longleftrightarrow \nabla(a, c)^{\boldsymbol{A}} \subseteq F \end{align*} $$
   is a partial order; and

2. (ii) for every $a \in A$ , $a \in F$ if and only if $\langle \boldsymbol {A}, \preccurlyeq _{F}^{\boldsymbol {A}} \rangle \vDash I(a)$ .

Finally, since $\vdash _{tr}$ is truth-equational, there is a set of equations $E(x)$ of $\vdash _{tr}$ such that for every $\langle \boldsymbol {A}, F \rangle \in \mathsf {Mod}^{\equiv }(\vdash _{tr})$ and $a \in A$

(34) $$ \begin{align} a \in F \Longleftrightarrow \boldsymbol{A} \vDash E(a). \end{align} $$

Now, we choose a theorem $\top $ of $\vdash _{tr}$ , and for every formula $\varphi \in \nabla (x, y)$ we consider the following basic operations of ${\vdash _{or}} \bigotimes {\vdash _{tr}}$ , in which $\pi _{i}$ is the projection map on the i-th coordinate: