2 Bochvar external logic

Kleene’s three-valued logics—introduced by Kleene in his Introduction to Metamathematics [Reference Kleene23]—are traditionally divided into two families, depending on the meaning given to the connectives: strong Kleene logics—counting strong Kleene and the logic of paradox—and weak Kleene logics, namely Bochvar logic [Reference Bochvar4] and paraconsistent weak Kleene logic (sometimes referred to as Hallden’s logic [Reference Halldén21]). Kleene logics are traditionally defined over the (algebraic) language of classical logic. However, the intent of one of the first developers of these formalisms, D. Bochvar, was to work within an enriched language allowing to express all classical “two-valued” formulas—which he referred to as external formulas—beside the genuinely “three-valued” ones.

The result of this choice is the language $\mathcal {L}\colon \langle \neg ,\lor ,\land ,J_{_0},J_{_1},J_{_2},0,1\rangle $ (of type $(1,2,2,1,1,1,0,0)$ ), which is obtained by enriching the classical language by three unary connectives $J_{_0},J_{_1},J_{_2}$ (and the constants $0,1$ ). The language $\mathcal {L}$ can be referred to as external language, in contrast with the traditional language upon which Kleene logics are defined. Let $\mathbf {Fm}$ refer to the formula algebra over the language $\mathcal {L}$ , and to $Fm$ as its universe.

The intended algebraic interpretation of the language $\mathcal {L}$ is traditionally given via the three-elements algebra displayed in Figure 1.

Figure 1 The algebra ${\mathbf {WK}}^{\mathrm {e}}$ .

The value is traditionally read as “meaningless” (see, e.g., [Reference Ferguson15, Reference Szmuc and Ferguson36]) due to its infectious behavior. It is immediate to check that the $\vee ,\land $ -reduct of ${\mathbf {WK}}^{\mathrm {e}}$ is not a lattice (it is an involutive bisemilattice), as it fails to satisfy absorption, hence the operations $\vee $ and $\land $ induce two (different) partial orders. In the following, we will refer to $\leq $ as the one induced by $\lor $ (i.e., $x\leq y$ iff $x\lor y = y$ ). With reference to such order, it holds .

The language $\mathcal {L}$ allows to define the so-called external formulas (see Definition 2.3), namely those that are evaluated into $\{0,1\}$ only (which is the universe of a Boolean subalgebra of ${\mathbf {WK}}^{\mathrm {e}}$ ), for any homomorphism $h\colon \mathbf {Fm}\to {\mathbf {WK}}^{\mathrm {e}}$ ( $J_{_k} \varphi $ , for any $\varphi \in Fm$ and $k\in \{0,1,2\}$ , are examples of external formulas).

In words, is the logic with the only distinguished value $1$ .Footnote ¹ is a linguistic expansion of Bochvar logic $\mathsf {B}$ , which is defined by the matrix ${\langle {\mathbf {WK}}, \{1\}\rangle }$ , where ${\mathbf {WK}}$ is the $J_{_k}$ -free reduct of ${\mathbf {WK}}^{\mathrm {e}}$ . Since is defined by a finite set of finite matrices, it is a finitary logic, in the sense that entails for some finite $\Delta \subseteq \Gamma $ (the relation among and other three-valued logics can be found in [Reference Ciucci and Dubois12]).

The following technicalities are needed to introduce a Hilbert-style axiomatization of .

The intuition behind the notion of external formulas is made precise by the following.

A Hilbert-style axiomatization of has been introduced by Finn and Grigolia [Reference Finn and Grigolia17]. In order to present it, let

and $\alpha ,\beta ,\gamma $ denote external formulas.

Axioms

1. (A1) $(\varphi \lor \varphi )\equiv \varphi $ ;

2. (A2) $(\varphi \lor \psi )\equiv (\psi \lor \varphi )$ ;

3. (A3) $((\varphi \lor \psi )\lor \chi )\equiv (\varphi \lor (\psi \lor \chi ))$ ;

4. (A4) $(\varphi \land (\psi \lor \chi )\equiv ((\varphi \land \psi )\lor (\varphi \land \chi ))$ ;

5. (A5) $\neg (\neg \varphi )\equiv \varphi $ ;

6. (A6) $\neg 1\equiv 0$ ;

7. (A7) $\neg ( \varphi \lor \psi )\equiv (\neg \varphi \land \neg \psi )$ ;

8. (A8) $0\vee \varphi \equiv \varphi $ ;

9. (A9) $J_{_2}\alpha \equiv \alpha $ ;

10. (A10) $J_{_0}\alpha \equiv \neg \alpha $ ;

11. (A11) $J_{_1}\alpha \equiv 0$ ;

12. (A12) $J_{_i}\neg \varphi \equiv J_{_{2-i}}\varphi $ , for any $i\in \{0,1,2\}$ ;

13. (A13) $ J_{_i}\varphi \equiv \neg (J_{_j}\varphi \vee J_{_k}\varphi )$ , with $i\neq j\neq k\neq i$ ;

14. (A14) $(J_{_i}\varphi \vee \neg J_{_i}\varphi )\equiv 1$ , with $i\in \{0,1,2\}$ ;

15. (A15) $((J_{_i}\varphi \vee J_{_k}\psi )\land J_{_i}\varphi )\equiv J_{_i}\varphi $ , with $i,k\in \{0,1,2\}$ ;

16. (A16) $(\varphi \lor J_{_i}\varphi )\equiv \varphi $ , with $i\in \{1,2\}$ ;

17. (A17) $J_{_0}(\varphi \lor \psi )\equiv J_{_0}\varphi \land J_{_0}\psi $ ;

18. (A18) $J_{_2}(\varphi \lor \psi )\equiv ( J_{_2}\varphi \land J_{_2}\psi )\vee ( J_{_2}\varphi \land J_{_2}\neg \psi )\vee ( J_{_2}\neg \varphi \land J_{_2}\psi ) $ ;

19. (A19) $\alpha \rightarrow (\beta \rightarrow \alpha ) $ ;

20. (A20) $( \alpha \rightarrow ( \beta \rightarrow \gamma ) ) \rightarrow (( \alpha \rightarrow \beta ) \rightarrow ( \alpha \rightarrow \gamma ))$ ;

21. (A21) $\alpha \wedge \beta \rightarrow \alpha $ ;

22. (A22) $\alpha \wedge \beta \rightarrow \beta $ ;

23. (A23) $\left ( \alpha \rightarrow \beta \right ) \rightarrow \left ( \left ( \alpha \rightarrow \gamma \right ) \rightarrow \left ( \alpha \rightarrow \beta \wedge \gamma \right ) \right ) $ ;

24. (A24) $\alpha \rightarrow \alpha \vee \beta $ ;

25. (A25) $\beta \rightarrow \alpha \vee \beta $ ;

26. (A26) $\left ( \alpha \rightarrow \gamma \right ) \rightarrow \left ( \left ( \beta \rightarrow \gamma \right ) \rightarrow \left ( \alpha \vee \beta \rightarrow \gamma \right ) \right ) $ ;

27. (A27) $\left ( \alpha \rightarrow \beta \right ) \rightarrow \left ( \left ( \alpha \rightarrow \lnot \beta \right ) \rightarrow \lnot \alpha \right ) $ ;

28. (A28) $\alpha \rightarrow \left ( \lnot \alpha \rightarrow \beta \right ) $ ;

29. (A29) $\lnot \lnot \alpha \rightarrow \alpha $ .

Deductive rule

Observe that the axiomatization contains a set of axioms (A19–A29), which, together with the rule of modus ponens, yields a complete axiomatization for classical logic (relative to external formulas).

The fact that coincides with the logic induced by the above introduced Hilbert-style axiomatization has been proved in [Reference Bonzio, Fano, Graziani and Pra Baldi6] (Finn and Grigolia [Reference Finn and Grigolia17, theorem 3.4] only proved a weak completeness theorem for ). We will henceforth indicate by both the consequence relation induced by the matrix ${\langle {\mathbf {WK}}^{\mathrm {e}},\{1\}\rangle }$ and the one induced by the above Hilbert-style axiomatization.

The logic is algebraizable with the quasivariety of Bochvar algebras ( $\mathsf {BCA}$ )—which will be properly introduced in the next section—as its equivalent algebraic semantics. This means that there exists maps $\tau \colon Fm\to \mathcal {P}(Eq)$ , $\rho \colon Eq\to \mathcal {P}(Fm)$ from formulas to sets of equations and from equations to sets of formulas such that

$$\begin{align*}\gamma_{1},\dots,\gamma_{n}\vdash_{\mathsf{B}_{e}}\varphi\iff\tau(\gamma_{1}),\dots,\tau(\gamma_{n})\vDash_{\mathsf{BCA}}\tau(\varphi)\end{align*}$$

and

The above conditions are verified by setting and (see [Reference Bonzio, Fano, Graziani and Pra Baldi6] for details). Moreover, Bochvar external logic enjoys the (global) deduction theorem, which we recall here.

3 Bochvar algebras and Płonka sums

We assume the reader has some familiarity with universal algebra and abstract algebraic logic (standard references are [Reference Bergman3, Reference Burris and Sankappanavar10] and [Reference Font18], respectively). In what follows, given a class of algebras $\mathsf {K}$ , the usual class-operator symbols $I(\mathsf {K}), S(\mathsf {K}),H(\mathsf {K}),P(\mathsf {K}),P_{u}(\mathsf {K})$ denote the closure of $\mathsf {K}$ under isomorphic copies, subalgebras, homomorphic images, products, and ultraproducts. A class of similar algebras $ \mathsf {K}$ is a quasivariety if $\mathsf {K}=ISPP_{u}(K)$ . It is a variety if is also closed under homomorphic images or, equivalently, if $\mathsf {K}=HSP(\mathsf {K})$ .

The class of Bochvar algebras, $\mathsf {BCA}$ for short, is introduced by Finn and Grigolia [Reference Finn and Grigolia17, pp. 233–234] as the algebraic counterpart for .

$\mathsf {BCA}$ forms a quasivariety which is not a variety [Reference Finn and Grigolia16, Reference Finn and Grigolia17], and it is generated by ${\mathbf {WK}}^{\mathrm {e}}$ , i.e., $\mathsf {BCA}=ISP(\mathbf {WK}^{e})$ . This is true in virtue of [Reference Czelakowski13, theorem 3.2.2], upon noticing that $\mathsf {BCA}$ algebraizes the logic , which is defined by the single matrix ${\langle {\mathbf {WK}}^{\mathrm {e}}, \{1\}\rangle }$ . The fact that ${\mathbf {WK}}^{\mathrm {e}}$ generates $\mathsf {BCA}$ was firstly stated by Finn and Grigolia [Reference Finn and Grigolia16, Reference Finn and Grigolia17]. Familiar examples of Bochvar algebras can be obtained by appropriately computing the external functions over a Boolean algebra, as indicates the following example.

For this reason, by $\mathbf {B}_{n}$ we will safely denote both the n-elements Boolean algebra and its $\mathsf {BCA}$ expansion obtained according to Example 3.2. Since ${\mathbf {WK}}^{\mathrm {e}}$ generates $\mathsf {BCA}$ , a quasi-equation holds in ${\mathbf {WK}}^{\mathrm {e}}$ if and only if it holds in every ${\mathbf A}\in \mathsf {BCA}$ .

The original quasi-equational basis for $\mathsf {BCA}$ , as provided in Definition 3.1, can be significantly enhanced by reducing the number of axioms and improving their intelligibility.Footnote ² It is known that the operations $J_{_{0}},J_{_{1}}$ can be defined as $J_{_2}\neg \varphi $ and $\neg (J_{_2}\varphi \lor J_{_2}\neg \varphi )$ , respectively. Thus, Bochvar algebras can be equivalently presented in the restricted language $\langle \vee ,\wedge , \neg , J_{_2}, 0,1\rangle $ , and this is particularly convenient for our next goal, namely to provide a new, simpler quasi-equational basis for $\mathsf {BCA}$ . This is accomplished in the next theorem.

The proof of the above Theorem 3.3 requires a significant amount of computations, which are included in the Appendix. Notice that, although $J_{_{0}},J_{_{1}}$ are definable from the remaining operations of $\mathsf {BCA}$ , a detailed investigation of the semantic properties of the full language significantly improve the logical and algebraic understanding of $\mathsf {BCA}$ : this is why in several subsequent parts of the paper we will explicitly refer to the full stock of operations.

We now introduce the variety of involutive bisemilattices, which plays a key role to understand the structure theory of Bochvar algebras.

The class of involutive bisemilattices forms a variety, which we denote by $\mathsf {IBSL}$ . $\mathsf {IBSL}$ is the so-called regularization of the variety of Boolean algebras: this is the variety satisfying all and only the regular identities that hold in Boolean algebras, namely those identities where exactly the same variables occur on both sides of the equality symbol. The variety $\mathsf {IBSL}$ is generated by the three element algebra ${\mathbf {WK}}$ , i.e., $\mathsf {IBSL}=HSP(\mathbf {WK})$ (see [Reference Bonzio, Paoli and Pra Bldi9, chap. 2], [Reference Bonzio, Gil-Férez, Paoli and Peruzzi7, Reference Płonka29]). It is not a direct consequence of Definition 3.1 (nor of Theorem 3.3) that the $J_{_k}$ -free reduct of any Bochvar algebra is an involutive bisemilattice: indeed Definition 3.1 implies that such reduct is a De Morgan bisemilattice (the regularization of de Morgan algebras). However, it is immediate to check that the identity I6 in Definition 3.4 holds in any Bochvar algebra, as it does in ${\mathbf {WK}}^{\mathrm {e}}$ . Although being the variety generated by the matrix that defines $\mathsf {B}$ , $\mathsf {IBSL}$ is not its algebraic counterpart, which rather consists of the proper quasivariety generated by ${\mathbf {WK}}$ . This quasivariety is called single-fixpoint involutive bisemilattices, $\mathsf {SIBSL}$ for short,Footnote ³ as its members are precisely the involutive bisemilattices with at most one fixpoint for negation, namely those containing at most one element a such that $a=\neg a$ .

The examples of Bochvar algebras we have made so far ( ${\mathbf {WK}}^{\mathrm {e}}$ and any Boolean algebra) consist of algebras having an $\mathsf {SIBSL}$ -reduct; this is actually true for any Bochvar algebra.

Proof. It suffices to check that every valid $\mathsf {SIBSL}$ -quasi-equation is also valid in $\mathsf {BCA}$ . To see this, let $\chi $ be a quasi-equation in the language of ${\mathbf {WK}}$ . We have that

$$ \begin{align*} \mathsf{SIBSL}\vDash\chi&\iff \\ {\mathbf{WK}}\vDash\chi&\iff \\ {\mathbf{WK}}^{\mathrm{e}}\vDash\chi&\iff \\ \mathsf{BCA}\vDash\chi. \end{align*} $$

The first equivalence holds because ${\mathbf {WK}}$ generates $\mathsf {SIBSL}$ , the second one because ${\mathbf {WK}}$ is the $\langle J_{_0},J_{_1},J_{_2}\rangle $ -free reduct of ${\mathbf {WK}}^{\mathrm {e}}$ and the last one because ${\mathbf {WK}}^{\mathrm {e}}$ generates $\mathsf {BCA}$ .

Clearly, since $\mathsf {SIBSL}\subset \mathsf {IBSL}$ , every Bochvar algebra has an $\mathsf {IBSL}$ -reduct. Although this fact, it is not the case that any (single-fixpoint) involutive bisemilattice can be turned into a Bochvar algebra: the reasons will be clear in the last part of the section.

The fact that the $J_{_k}$ -free reduct of every Bochvar algebra is an involutive bisemilattice (Proposition 3.5) carries to the relevant observation that any such reduct can be represented as a Płonka sum of Boolean algebras [Reference Bonzio, Gil-Férez, Paoli and Peruzzi7]. Płonka sums are general constructions introduced by the polish mathematician J. Płonka [Reference Płonka27, Reference Płonka28, Reference Płonka30] (more comprehensive expositions are [Reference Bonzio, Paoli and Pra Bldi9, chap. 2], [Reference Płonka, Romanowska, Romanowska and Smith31, Reference Romanowska and Smith33])—and now going under his name. In brief, the construction consists of “summing up” similar algebras, organized into a semilattice direct system and connected via homomorphisms, into a new algebra. In more details, the (semilattice direct) system is formed by a family of similar algebras $\{{\mathbf A}_{i}\}_{i\in I}$ with disjoint universes, such that the index set I forms a lower-bounded semilattice $(I, \vee , i_{0})$ —we denote by $\leq $ the induced partial order—and, moreover, is made of a family $\{p_{ij}\}_{i\leq j}$ of homomorphisms $p_{ij}\colon {\mathbf A}_{i}\to {\mathbf A}_j$ , defined from the algebra ${\mathbf A}_i$ to the algebra ${\mathbf A}_j$ , whenever $i\leq j$ , for $i,j \in I$ . Such homomorphisms satisfy a further compatibility property: $p_{ii}$ is the identity, for every $i\in I$ and $p_{jk}\circ p_{ij} = p_{ik}$ , for every $i\leq j\leq k$ .

Given a semilattice direct system of algebras ${\langle \{{\mathbf A}_i\}_{i\in I}, (I, \vee , i_{0}), \{p_{ij}\}_{i\leq j}\rangle }$ , the Płonka sum over it is the new algebra ${\mathbf A}={\mathcal {P}}_{\unicode{x0142}}({\mathbf A}_{i})_{i\in I}$ (of the same similarity type of the algebras $\{{\mathbf A}_i\}_{i\in I}$ ) whose universe is the union $A=\displaystyle \bigcup _{i\in I} A_i $ and whose generic n-ary operation g is defined as

(1)

where $k= i_{1}\vee \dots \vee i_{n}$ and $a_1\in A_{i_1},\dots , a_{n}\in A_{i_n}$ . If the similarity type contains any constant operation e, then $e^{{\mathbf A}} = e^{{\mathbf A}_{i_0}}$ . The algebras $\{{\mathbf A}_i\}_{i\in I}$ are called the fibers of the Płonka sum. A fiber $\mathbf {A}_{i}$ is trivial if its universe is a singleton. As already stated above, an element a is a fixpoint when $a=\neg a$ . Equivalently, using the Płonka sum representation, a fixpoint can be understood as the universe of a trivial fiber.

With this terminology at hand, a remarkable result states that any member of the variety $\mathsf {IBSL}$ of involutive bisemilattices is isomorphic to the Płonka sum over a semilattice direct system of Boolean algebras (see [Reference Bonzio, Gil-Férez, Paoli and Peruzzi7, Reference Bonzio, Paoli and Pra Bldi9]). In a Płonka sum of Boolean algebras ${\mathbf A}={\mathcal {P}}_{\unicode{x0142}}({\mathbf A}_{i})_{i\in I}$ , a key role is played by the absorption function $\varphi \land (\varphi \lor \psi )$ , in the following sense. Given $a,b\in A$ , it is possible to check that $a,b\in A_{i}$ for some $i\in I$ if and only if $a\land (a\lor b)=a$ and $b\land (b\lor a)=b$ . Moreover, if $a\in A_{i}$ and $i\leq j$ , it holds $p_{ij}(a)=a\land (a\lor b)$ , for any $b\in A_{j}$ . In the light of the above observations, given a Bochvar algebra ${\mathbf A}$ we will denote by ${\mathcal {P}}_{\unicode{x0142}}({\mathbf A}_{i})_{i\in I}$ the Płonka decomposition of its $\mathsf {IBSL}$ -reduct.

It shall be clear to the reader that a Bochvar algebra can not be represented as a Płonka sum (of some class of algebras) in the usual sense (recalled above). The reason is that the operations $J_{_k} $ , for any $k\in \{0,1,2\}$ , are not computed according to condition (1). Indeed, if ${\mathcal {P}}_{\unicode{x0142}}({\mathbf A}_{i})_{i\in I}$ is a Płonka sum and $a\in A_{i}$ , Axiom (11) entails that $J_{k}a\in A_{i_0}$ (with $i_0$ the least element of the index set I), while condition (1) requires that $J_{k}a\in A_{i}$ . Nonetheless, the fact that any Bochvar algebra has an $\mathsf {IBSL}$ -reduct (Proposition 3) suggests that Płonka sums are good candidates to provide a representation theorem. Indeed we will rely on the Płonka sum representation of the $\mathsf {IBSL}$ -reduct of a Bochvar algebra to “reconstruct” the additional operations $J_{_k} $ and provide a (unique) Płonka sum decomposition for any Bochvar algebra. We begin by studying the behavior of the maps $J_{_k} $ with respect to the $\mathsf {IBSL}$ -reduct of a Bochvar algebra.

Proof. It suffices to notice that (1) holds if and only if

$$\begin{align*}\mathsf{BCA}\vDash \varphi\land 0\thickapprox 0\Rightarrow J_{_1}\varphi\thickapprox 0\end{align*}$$

and this quasi-equation clearly holds in ${\mathbf {WK}}^{\mathrm {e}}$ . The same applies to (2), (3) with respect to the quasi-equations $\varphi \land 0\thickapprox 0\Rightarrow J_{_2}\varphi \thickapprox \varphi $ , $ \varphi \land 0\thickapprox 0\Rightarrow J_{_1}\varphi \thickapprox \neg \varphi $ .

Observe that, when one takes into account the whole algebra ${\mathbf A}$ , $J_{_1}$ in general does not coincide with the constant function $0$ , as in ${\mathbf {WK}}^{\mathrm {e}}$ it holds .

The next result summarizes the key features of the operation $J_{_1}$ .

Proof. (1). For every $a,b\in A_{i}$ , $a\land (a\lor b)=a$ and $b\land (b\lor a)=a$ . Since

$$\begin{align*}{\mathbf{WK}}^{\mathrm{e}}\vDash \varphi\land(\varphi\lor \psi)\thickapprox\varphi \ \& \ \psi\land(\psi\lor \varphi)\thickapprox\psi\Rightarrow J_{_1}\varphi\thickapprox J_{_1}\psi, \end{align*}$$

it follows $J_{_1}a=J_{_1}b$ .

(2). Follows from (1) in Lemma 3.6.

(3). Holds because ${\mathbf {WK}}^{\mathrm {e}}\vDash \varphi \thickapprox \neg \varphi \Rightarrow J_{_1}\varphi \thickapprox 1$ .

(4). For $a\in A_{i}$ , it holds $p_{i_{0}i}(J_{_1}a)=J_{_1}a\land (J_{_1}a\lor a)$ and observe that

$$\begin{align*}{\mathbf{WK}}^{\mathrm{e}}\vDash J_{_1}\varphi\land(J_{_1}\varphi\lor \varphi)=\varphi\land\neg\varphi,\end{align*}$$

so $p_{i_{0}i}(J_{_1}a)=J_{_1}a\land (J_{_1}a\lor a)=a\land \neg a$ .

As a notational convention, let us denote by $1_i$ and $0_i$ the top and the bottom elements, respectively, of a generic fiber ${\mathbf A}_i$ in a Płonka sum of Boolean algebras.

Proof. Clearly ${\mathbf {WK}}^{\mathrm {e}}\vDash J_{_2}\varphi \land (J_{_2}\varphi \lor \varphi )\thickapprox \varphi $ , and ${\mathbf {WK}}^{\mathrm {e}}\vDash J_{_0}\varphi \land (J_{_0}\varphi \lor \varphi )\thickapprox \neg \varphi $ , so, for $a\in A_{i}$ , $p_{i_{0}i}(J_{_2}a)=J_{_2}a\land (J_{_2}a\lor a)=a$ and $p_{i_{0}i}(J_{_0}a)=\neg a$ . This proves (3) and that $p_{i_{0}i}$ is surjective, for every $i\in I$ . Let now $i\leq j$ . Since $p_{i_{0}j}=p_{ij}\circ p_{i_{0}i}$ is surjective, also $p_{ij}$ is surjective. This shows that ${\mathcal {P}}_{\unicode{x0142}}({\mathbf A}_{i})_{i\in I}$ has surjective homomorphisms (1). (4) holds as it is equivalent to the quasi-equation

$$\begin{align*}\varphi\land(\varphi\lor\psi)\thickapprox\varphi \ \& \ \psi\land(\psi\lor\varphi)\thickapprox\psi\Rightarrow J_{_2}(\varphi\lor\psi)\thickapprox J_{_2}\varphi\lor J_{_2}\psi,\end{align*}$$

which is true in ${\mathbf {WK}}^{\mathrm {e}}$ .

(2) Suppose, by contradiction, that there is $j\in I$ such that $j\neq i_{0}$ and $p_{i_{0} j}$ is an injective homomorphism. Thus, by Lemma 3.7 and the fact that $J_{_2}(\varphi \land \neg \varphi )\thickapprox 0$ is true in ${\mathbf {WK}}^{\mathrm {e}}$ , it holds $J_{_1} 0_j = 0$ and $J_{_2} 0_j = 0$ . Moreover, by (12), $J_{_0} 0_j = 1$ , hence $J_{_i} 0 = J_{_i} 0_{j}$ , for every $i\in \{0,1,2\}$ . Therefore, by the quasi-equation (13), $0 = 0_{j}$ , a contradiction.

Recall that, for any non-trivial Boolean algebra ${\mathbf A} = {\langle A,\wedge ,\vee , \neg , 0,1 \rangle }$ and any $a\in A$ , one can turn the interval $[0,a] = \{x\in A\;|\; x\leq a\}$ into a Boolean algebra $\mathbf {[0,a]} = {\langle [0,a], \wedge , \vee , ^{\ast }, 0, a\rangle } $ , where $x^{\ast } = \neg x\wedge a$ . We will refer to such an algebra as an interval Boolean algebra. In the following result we show that any Boolean algebra in the Płonka sum representation of the $\mathsf {IBSL}$ -reduct of a Bochvar algebra is isomorphic to a specific interval Boolean algebra in the lowest fiber.

Combining Proposition 3.12 and Remark 3.13 we get that, for any $i\in I \mathbf {[0,\top ]}\cong {\mathbf A}_{i_{0}}/Ker(p_{i_{0}i})$ . In the following we use the interval characterization proved in Proposition 3.12 to establish some properties that will be used in the subsequent sections.

Proof. (1). If $1_{i}<1_{j}$ then $1_{i}\land 1_{j}=1_{j}\neq 1_{i}$ . The following two quasi-equations hold in ${\mathbf {WK}}^{\mathrm {e}}$ :

$$ \begin{align*} \varphi\lor\neg\varphi\leq\psi\lor\neg\psi\Rightarrow J_{_2}(\psi\lor\neg\psi)\leq J_{_2}(\varphi\lor\neg\varphi), \\ J_{_2}(\varphi\lor\neg\varphi)\thickapprox J_{_2}(\psi\lor\neg\psi)\Rightarrow \varphi\lor\neg\varphi\thickapprox\psi\lor\neg\psi. \end{align*} $$

From the former we have that $J_{_2}1_{j}\leq J_{_2}1_{i}$ and, since $1_i\neq 1_j$ , from the latter we conclude $J_{_2}1_{j}<J_{_2}1_{i}$ . This entails $J_{_2}({\mathbf A}_j) = [0,J_{_2} 1_j]\subset [0,J_{_2} 1_i] = J_{_2}({\mathbf A}_i)$ .

(2) is equivalent to the equation $J_{_2}(\varphi \land (\varphi \lor \psi ))\leq J_{_2}\varphi $ , which is true in ${\mathbf {WK}}^{\mathrm {e}}$ .

The following result provides necessary conditions for an $\mathsf {SIBSL}$ to be the reduct of a Bochvar algebra.

Proof. Let ${\mathbf A}\in \mathsf {BCA}$ . By Proposition 3, the $J_{_k}$ -free reduct (with $k\in \{0,1,2\}$ ) of ${\mathbf A}$ is a single-fixpoint involutive bisemilattice, thus it is isomorphic to a Płonka sum over a semilattice direct systems ${\langle \{{\mathbf A}_i\}_{i\in I}, (I,\leq ), p_{ij}\rangle }$ of Boolean algebras (see [Reference Bonzio, Gil-Férez, Paoli and Peruzzi7, Reference Bonzio, Paoli and Pra Bldi9]), whose homomorphisms are surjective and $p_{i_{0}i}$ is not injective (by Lemma 3.9). Moreover, for every $i\in I$ , $J_{_2} 1_{i}$ is the element in ${\mathbf A}_{i_{0}}$ such that ${\mathbf A}_{i}\cong \mathbf {[0, J_{_2} 1_{i}]}$ , by Proposition 3.12 (which also ensures the isomorphisms are given by $J_{_2}$ and $p_{i_{0}i}$ ). Lemma 3.14 ensures that, if $i<j$ then $J_{_2}1_{j}<J_{_2}1_{i}$ . Finally, notice that $J_{_2}(\varphi \lor \neg \varphi )\thickapprox J_{_2}(\psi \lor \neg \psi )\Rightarrow \varphi \lor \neg \varphi \thickapprox \psi \lor \neg \psi $ holds in ${\mathbf {WK}}^{\mathrm {e}}$ . Therefore, since $i\neq j$ entails $1_{i}\neq 1_{j}$ , we conclude $a_{i}=J_{_2}1_{i}\neq J_{_2}1_{j}=a_{j}$ .

We now show that the decomposition is unique up to isomorphism, namely that different choices of the element $J_{_2} a\in p_{i_{0}i}^{-1}(a)$ (for any $a\in A_{i}$ ) on isomorphic $\mathsf {IBSL}$ -reducts lead to isomorphic Bochvar algebras. Observe that the decomposition of the $\mathsf {IBSL}$ -reduct of ${\mathbf A}$ is unique up to isomorphism (see [Reference Bonzio, Gil-Férez, Paoli and Peruzzi7, Reference Bonzio, Paoli and Pra Bldi9]). So, suppose that ${\mathbf A}$ and $\mathbf {B}$ are Bochvar algebras whose $\mathsf {IBSL}$ -reducts ${\mathbf A}'$ , $\mathbf {B}'$ are isomorphic via an isomorphism $\Phi \colon {\mathbf A}'\to \mathbf {B}'$ . We claim that ${\mathbf A}\cong \mathbf {B}$ via $\Phi $ . To this end, we want to show that $\Phi $ preserves also the operations $J_{_{k}}$ , for any $k\in \{0,1,2\}$ . Recall from the theory of Płonka sums that $\Phi $ preserves the fibers (details can be found in [Reference Bonzio5, Reference Bonzio, Loi and Peruzzi8]) in the following sense: for any fiber ${\mathbf A}_i$ in the Płonka sum decomposition of ${\mathbf A}'$ , $\Phi ({\mathbf A}_i) \cong \mathbf {B}_{\varphi (i)}$ , where $\varphi $ is the isomorphism induced by $\Phi $ on the semilattice of indexes and $\mathbf {B}_{\varphi (i)}$ is a fiber in the Płonka sum decomposition of $\mathbf {B}' $ . In particular, the following diagram commutes, for any $i\in I$ (with a slight abuse of notation we indicate $\varphi (i_0)$ with $i_0$ as it is still a lower bound in $\varphi (I)$ ).

We claim that ${\mathbf A}\cong \mathbf {B}$ via $\Phi $ . Let $a\in A$ ; in particular, $a\in A_i$ , for some $i\in I$ . By Lemma 3.9, $J_{_2}^{{\mathbf A}} a \in p_{i_{0}i}^{-1}(a)$ and $J_{_2}^{\mathbf {B}}\Phi (a)\in q_{i_{0}\varphi (i)}^{-1}(\Phi (a))$ , hence, by the commutativity of the above diagram, the fact that $J_{_2}$ is the isomorphism between ${\mathbf A}_{i}$ and $ {\mathbf A}_{i_{0}}/Ker(p_{i_{0}i}) $ (Remark 3.13) and that $\Phi _{\restriction {{\mathbf A}_{i}}}$ and $\Phi _{\restriction {{\mathbf A}_{i_{0}}}}$ are isomorphisms) follows that $\Phi (J_{_2}^{{\mathbf A}} a) = J_{_2}^{\mathbf {B}}\Phi (a)$ . Therefore, we have that $\Phi $ is a homomorphism with respect to $J_{_2}$ (and hence with respect of $J_{_1}$ and $J_{_0}$ , which can be defined in term of $J_{_2}$ ) and this shows that ${\mathbf A}\cong \mathbf {B}$ , namely the Płonka sum decomposition is unique up to isomorphism.

We now show that the conditions displayed in the above theorem are also sufficient to equip any $\mathsf {SIBSL}$ with a $\mathsf {BCA}$ -structure.

The first part of condition (2) in Theorems 3.15 and 3.17 can be replaced by the assumption that $1/\operatorname {\mathrm {Ker}}_{p_{i_{0}i}}$ is a principal filter, for every $i\in I$ . Therefore, we obtain the following.

We conclude this section with an example which empathizes the role of condition (2) in Corollary 3.18 and shows an $\mathsf {SIBSL}$ that can not be turned into a Bochvar algebra.

The results included in the subsequent sections will strongly make use of the Płonka sum decomposition of a Bochvar algebra provided in Theorem 3.15. In the exposition of many results we will take for granted some of the details introduced in this section.

4 On quasivarieties of Bochvar algebras

The logic is algebraizable with respect to the quasivariety of Bochvar algebras and it is well known that there exists a dual isomorphism between the lattice of extensions of and the lattice of subquasivarieties of $\mathsf {BCA}$ . In this section we characterize such lattices, proving that they consist of the three-elements chain. In order to do so, we take advantage of the recent results in [Reference Moraschini, Raftery and Wannenburg25], therefore applying general properties of passive structurally complete (PSC) quasivarieties. PSC is a weakened variant of structural completeness (see, among others, [Reference Campercholi, Stronkowski and Vaggione11, Reference Dzik and Stronkowski14, Reference Raftery32, Reference Rybakov34]), a notion defined, in its algebraic version, as follows.

Definition 4.1. A quasivariety $\mathsf {K}$ is structurally complete (SC) if, for every quasivariety  $\mathsf {K}^{\prime }$ :

$$\begin{align*}\mathsf{K}^{\prime}\subsetneq\mathsf{K}\;\Rightarrow\;\mathbb{V}(\mathsf{K}^{\prime})\subsetneq\mathbb{V}(\mathsf{K}), \end{align*}$$

where $\mathbb {V}(\mathsf {K})=HSP(\mathsf {K})$ is the variety generated by $\mathsf {K}$ . Recall that a quasi-identity $ \varphi _{1}\thickapprox \psi _{1}\;\&\dots \&\;\varphi _{n}\thickapprox \psi _{n}\Rightarrow \varphi \thickapprox \psi $ is passive in a quasivariety $\mathsf {K}$ if, for every substitution $h\colon \mathbf {Fm}\to \mathbf {Fm}$ , there exists an algebra ${\mathbf A}\in \mathsf {K}$ such that ${\mathbf A}\nvDash h(\varphi _{i})\thickapprox h(\psi _{i})$ for some $1\leq i\leq n$ . Put differently, a quasi-identity is passive if it is equivalent to a quasi-identity whose conclusion is $x\thickapprox y$ , where $x,y$ are variables that do not appear in the antecedent (see [Reference Aglianò and Ugolini1, sec. 4.3]) for details.Footnote ⁴ We take the following as a definition of passive structural completeness (PSC).

For a quasivariety $\mathsf {K}$ , one of the consequences of being PSC amounts to be generated by a single algebra, i.e., to be singly generated (see [Reference Moraschini, Raftery and Wannenburg25, sec. 7]). Recall that an algebra ${\mathbf A}$ is a retract of $\mathbf {B}$ if there exist two homomorphisms $\iota \colon {\mathbf A}\to \mathbf {B}$ and $r\colon \mathbf {B}\to {\mathbf A}$ such that $r\circ \iota $ is the identity map on ${\mathbf A}$ . This forces r to be surjective and $\iota $ to be injective: we will call r a retraction. ${\mathbf A}$ is a common retract of a quasivariety $\mathsf {K}$ if it is a retract of every non-trivial member of $\mathsf {K}$ . We denote by $Ret(\mathsf {K},{\mathbf A})=\{\mathbf {B}\in \mathsf {K}: {\mathbf A} \ \text {is a retract of}\ \mathbf {B}\}$ the members of $\mathsf {K}$ having ${\mathbf A}$ as a retract.

The main tool that will be instrumental for our purposes is the following.

The following summarizes the properties of the quasivariety $\mathsf {BCA}$ with respect to the above introduced notions.

Proof. $\mathrm{(i)}$ . The quasi-identity

(NF) $$\begin{align} J_{_1}\varphi\thickapprox 1\Rightarrow \psi\thickapprox 1 \end{align}$$

is passive in $\mathsf {BCA}$ . Indeed, for each substitution, the antecedent $J_{_1} \varphi \thickapprox 1$ is falsified in every Bochvar algebra containing no trivial algebra in its Płonka sum decomposition. On the other hand, it is immediate to check that $\mathsf {BCA}\nvDash (NF)$ .

$\mathrm{(ii)}$ Is straightforward, as every Boolean algebra is also a Bochvar algebra (see Example 3.2), and $\mathbf {B}_{2}$ is the $0$ -generated Boolean algebra.

$\mathrm{(iii)}$ Follows from $(ii)$ and Theorem 4.3.

It is easy to check that, in $\mathsf {BCA}$ , $(NF)$ is equivalent to the quasi-identity

$$ \begin{align*}\varphi\thickapprox\neg \varphi\;\Rightarrow\; \psi\thickapprox \delta.\end{align*} $$

Since $(NF)$ is valid in the quasivariety $Ret(\mathsf {BCA},\mathbf {B}_{2})$ , it demands that the underlying involutive bisemilattice of any of its non-trivial members lacks trivial fibers. A stronger fact is established in the following lemma.

Proof. $(\Rightarrow )$ . By Proposition 4.4, $Ret(\mathsf {BCA}, \mathbf {B}_{2})$ is a (maximal) PSC quasivariety and $(NF)$ is a passive quasi-identity, therefore $Ret(\mathsf {BCA},\mathbf {B}_{2})\vDash (NF)$ . $(\Leftarrow )$ . Suppose ${\mathbf A}\vDash (NF)$ , therefore its $\mathsf {IBSL}$ -reduct lacks trivial fibers. If ${\mathbf A}$ is a Boolean algebra, the conclusion immediately follows. Differently, $\lvert I\rvert \; \geq 2$ . Let $i>i_{0}$ and consider the set $X=\{a\in A_{i_{0}}\;|\; p_{i_{0}i} (a) =1_{i}\}$ ; let $F_0$ be an ultrafilter on ${\mathbf A}_{0}$ containing X (it exists since the filter generated by X is proper, as the Płonka decomposition lacks trivial algebras). Since all the homomorphisms between fibers of ${\mathbf A}$ are surjective and ${\mathbf A}$ lacks trivial fibers, $F_{j}=p_{i_{0}j}[F_{0}]$ is an ultrafilter on ${\mathbf A}_{j}$ , for each $j\in I$ (this readily follows from the fact that $F_0$ is an ultrafilter extending X). Set $F=\displaystyle \bigcup _{k\in I}p_{i_{0}k}[F_{0}]$ and let $r\colon {\mathbf A}\to \mathbf {B}_{2}$ be defined, for each $a\in A$ , as

$$\begin{align*}r(a) = \begin{cases} 1, \text{ if } a\in F, \\ 0, \text{ otherwise}. \end{cases} \end{align*}$$

We want to show that r is a retraction. We show that r is compatible with $\land $ , $ \neg $ and $J_{_2}$ . Let $a\in A_{i}$ , $b\in A_{j}$ and set $k=i\lor j$ : we have $r(a\land b)=1\iff a\land b\in F_{k}\iff p_{ik}(a),p_{jk}(b)\in F_{k}$ . Suppose, by contradiction, $a\notin F_{i}$ . Then $p^{-1}_{i_{0}i}(a)\notin F_{0}$ which entails $p_{i_{0}k}(p^{-1}_{i_{0}i}(a))\notin F_{k}$ . However, by the composition property of Płonka homomorphisms, $p_{i_{0}k}(p^{-1}_{i_{0}i}(a))= p_{ik}(p_{i_{0}i}(p^{-1}_{i_{0}i}(a))=p_{ik}(a)\in F_{k}$ , a contradiction. So $a\in F_{i}$ . The same argument applies to b, and we conclude $b\in F_{j}$ . Therefore $r(a)\land r(b)=1=r(a\land b)$ .

Let $r(a) = 1$ , for some $a\in A$ , then $a \in F_j$ , for some $j\in I$ and, since $F_j$ is an ultrafilter on ${\mathbf A}_j$ , $\neg a\not \in F_j$ , thus $r(\neg a) = 0 = \neg r(a)$ .

Finally, we show the compatibility of r with $J_{_2}$ . For any $a\in A_{j}$ , we have $r(J_{_2}a)=1\iff J_{_2}a\in F_{0}\iff p_{i_{0}i}(J_{_2}a)=a\in F_{i}\iff J_{_2}r(a)=1$ . So r is a surjective homomorphism. Setting $\iota \colon \mathbf {B}_{2}\to {\mathbf A}$ such that $\iota (1)=1, \iota (0)=0$ , we have that r is a retraction. This shows $\mathbf {B}_{2}$ is a retract of ${\mathbf A}$ , so ${\mathbf A}\in Ret(\mathsf {BCA},\mathbf {B}_{2})$ .

In analogy with the terminology introduced in [Reference Paoli and Pra Baldi26], we call the quasivariety $Ret(\mathsf {BCA},\mathbf {B}_{2})$ : nonparaconsistent Bochvar algebras, $\mathsf {NBCA}$ in brief. Since $\mathsf {NBCA}$ is a (maximal) PSC subquasivariety of $\mathsf {BCA}$ , we know that $\mathsf {NBCA}$ , as a quasivariety, is generated by a single algebra $\mathbf {A}$ , namely $\mathsf {NBCA}=ISPP_{u}(\mathbf {A})=ISP(\mathbf {A})$ (see [Reference Moraschini, Raftery and Wannenburg25, theorem 4.3]). We now introduce an example of a Bochvar algebra which will play an important role.

Example 4.7 ( $\mathbf {B}_{4}\oplus \mathbf {B}_{2}$ ).

Let $\mathbf {B}_{4}\oplus \mathbf {B}_{2}$ denote the involutive bisemilattice whose Płonka sum consists of a system made of the four-element Boolean algebra $\mathbf {B}_{4}$ , the two-element one $\mathbf {B}_{2}$ , the two-element lattice as index set as in the following diagram (where the arrows stand for the homomorphism $p_{i_{0}i}\colon \mathbf {B}_{4}\to \mathbf {B}_{2}$ ).

It follows from the structure theory developed in Section 3 that the unique way to turn $\mathbf {B}_{4}\oplus \mathbf {B}_{2}$ into a Bochvar algebra is by defining $J_{_2} \top = a$ , $J_{_2}\perp = 0$ , $J_{_1} \top = J_{_1}\perp = \neg a$ and $J_{_0}\top = 0$ , $J_{_0}\perp = a$ (recall that $J_{_2}$ is the identity on $\mathbf {B}_4$ , $J_{_1}$ is the constant onto $0$ and $J_{_0}$ is negation). With a slight abuse of notation we will indicate this unique Bochvar algebra by $\mathbf {B}_{4}\oplus \mathbf {B}_{2}$ , as its $\mathsf {IBSL}$ -reduct.

Upon noticing that every non-trivial $\mathsf {NBCA}$ lacks trivial subalgebras (this amounts to say that $\mathsf {NBCA}$ is Kollár), [Reference Moraschini, Raftery and Wannenburg25, corollary 7.8] ensures that $\mathbf {B}_{2}$ is the unique relatively simple member of $\mathsf {NBCA}$ , namely it is the unique algebra in the quasivariety whose lattice of relative congruences is a two-element chain. Moreover, $\mathbf {B}_{4}\oplus \mathbf {B}_{2}$ is a relatively subdirectly irreducible member of $\mathsf {NBCA}$ which is therefore not simple. In other words, $\mathsf {NBCA}$ is not relatively semisimple, unlike $\mathsf {BCA}$ .

Observe that any Bochvar algebra satisfies the absorption law $\varphi \thickapprox \varphi \land (\varphi \lor \psi )$ if and only if its involutive bisemilattice reduct is a Boolean algebra. We call $\mathsf {JBA}$ the quasivariety axiomatized by adding the absorption law to the quasi-equational theory of $\mathsf {BCA}$ . It is immediate to verify that $\mathsf {JBA}$ and $\mathsf {BA}$ are term equivalent by interpreting the operation $J_{_2}$ as the identity map. The next theorem characterizes the structure of the lattice of non-trivial subquasivarieties of $\mathsf {BCA}$ , proving that it consists of the following three-elements chain.

Moreover, from the fact that $\mathsf {BCA}$ is not SC we can infer the following.

Let now switch our attention to the logical setting, relying on the bridge results connecting an algebraizable logic (and its extensions) with its algebraic counterpart(s). Let be the logic obtained by adding to the rule

(EFJ) $$\begin{align} J_{_1}\varphi\vdash \psi. \end{align}$$

This logic is a proper extension of , as $(EFJ)$ is the logical pre-image of $(NF)$ via the transformer formula-equations transformer $\tau $ and $(NF)$ is not valid in (a counterexample is easily found in ${\mathbf {WK}}^{\mathrm {e}}$ ). In the light of Theorem 4.9 we obtain the following.

In a logical perspective, the notions of PSC and SC have been investigated in several contributions, such as [Reference Bergman2, Reference Raftery32, Reference Rybakov34, Reference Wroński38]. For a logic $\vdash $ , being SC amounts to the fact that each admissible rule is derivable in $\vdash $ . In other words, $\vdash $ is SC if for every rule (R) of the form $\langle \Gamma ,\psi \rangle $ :

$$\begin{align*}(\vdash\varphi\iff\vdash_{R}\varphi)\Rightarrow \Gamma\vdash\psi,\end{align*}$$

where $\vdash _{R}$ is the extension of $\vdash $ obtained by adding (R) to $\vdash $ . Clearly, the converse implication in the above display is always true. A passive rule is of the form $\langle \Gamma , y\rangle $ , where no member of $\Gamma $ contains occurrences of y, namely $y\notin \textit{Var}(\Gamma )$ . Accordingly, we say that a logic $\vdash $ is PSC if every passive, admissible rule is derivable.Footnote ⁶

The following corollary emphasizes the logical meaning of the previous results on the subquasivarieties of $\mathsf {BCA}$ .

5 Amalgamation in quasivarieties of Bochvar algebras

In the context of algebraizable logics, several logical properties can be established by means of the so-called bridge theorems, whose general form is

where the quasivariety $\mathsf {K}$ is the equivalent algebraic semantics of . A valid instance of the above equivalence can be obtained by replacing P with “Deduction theorem” and Q with “Equationally definable principal relative congruences” (see [Reference Czelakowski13, theorem Q.9.3]).

In the light of the results of Section 4, $\mathsf {BCA}$ and $\mathsf {NBCA}$ are the only interesting quasivarieties of Bochvar algebras. In this section we show that they enjoy the amalgamation property (AP) or, equivalently, that their associated logics enjoy the Craig interpolation property. The strategy for proving (AP) for $\mathsf {BCA}$ consists in providing a sufficient condition implying (AP), established in [Reference Metcalfe, Montagna and Tsinakis24, theorem 9] for varieties, and that naturally extends to quasivarieties (Theorem 5.2).

Recall that a V-formation (see Figure 2) is a $5$ -tuple $\left ( \mathbf {A},\mathbf {B},{\mathbf C},i,j\right ) $ such that $\mathbf {A,B,C}$ are similar algebras, and $i\colon \mathbf {A\rightarrow B},j\colon \mathbf {A\rightarrow C}$ are embeddings. A class $\mathsf {K}$ of similar algebras is said to have the amalgamation property if for every V-formation with $\mathbf {A},\mathbf {B},{\mathbf C}\in \mathsf {K}$ there exists an algebra $\mathbf {D}\in \mathsf {K}$ and embeddings $h\colon \mathbf {B} \mathbf {\rightarrow D},k\colon {\mathbf C}\mathbf {\rightarrow D}$ such that $k\circ j=h\circ i$ . In such a case, we also say that $({\mathbf D}, h, k)$ is an amalgam of the V-formation $\left ( \mathbf {A},\mathbf {B} ,{\mathbf C},i,j\right ) $ .

Figure 2 A generic amalgamation schema.

The following lemma is originally due to Grätzer [Reference Grätzer20] (it can be also found in [Reference Metcalfe, Montagna and Tsinakis24]), while the subsequent theorem is the obvious adaptation to quasivarieties of a theorem by Metcalfe, Montagna, and Tsinakis [Reference Metcalfe, Montagna and Tsinakis24, theorem 9]. We insert the proofs for the completeness of the exposition.

The following provides a sufficient condition for a quasivariety to have the (AP) and it reduces somehow the search for an amalgam to a subclass of a quasivariety. As a notational convention, by $Co_{\mathsf {K}}^{{\mathbf A}}$ we denote the lattice of $\mathsf {K}$ -congruences on an algebra ${\mathbf A}$ , namely the congruences $\theta $ on $\mathbf {A}$ such that $\mathbf {A}/\theta \in \mathsf {K}$ . Let $\{\theta _{i}\}_{i\in I}$ be a family of $\mathsf {K}$ -congruences on an algebra $\mathbf {A}\in \mathsf {K}$ . We say that $\mathbf {A}$ is subdirectly irreducible relative to $\mathsf {K}$ , or just relatively subdirectly irreducible, when $\bigwedge _{i\in I}\theta _{i}=\Delta ^{\mathbf {A}}$ entails $\theta _{i}=\Delta ^{\mathbf {A}}$ for some $i\in I$ . Moreover, given a quasivariety $\mathsf {K}$ , by $\mathsf {K}_{RSI}$ we indicate the class of relatively subdirectly irreducible members of $\mathsf {K}$ . If $\mathsf {K}$ is a variety, we simply write $\mathsf {K}_{SI}$ .

We conclude this section by proving that also the quasivariety $\mathsf {NBCA}$ has the (AP). Before proceeding further, it is worth noticing that we cannot apply the same strategy used in the case of $\mathsf {BCA}$ . This is a consequence of the following remark, which also proves that the logic $\mathsf {NB}_{e}$ does not have a local deduction theorem.

Nonetheless, (AP) holds for $\mathsf {NBCA}$ , as shown in the following.

Proof. Let $({\mathbf A},\mathbf {B},{\mathbf C}, f, g)$ be a V-formation in $\mathsf {NBCA}$ . By Theorem 5.4, there exists an amalgam $(h,k,\mathbf {D})$ with ${\mathbf D}\in \mathsf {BCA}$ . If ${\mathbf D}$ has no trivial fibers, then $(h,k,\mathbf {D})$ is also an amalgam in $\mathsf {NBCA}$ . Otherwise, let $u\in I$ be the index of the trivial fiber $\mathbf {D}_{u}$ with universe $\{u\}$ , where I is the underlying semilattice of $\mathbf {D}$ and homomorphisms $p_{ij}$ for every $i\leq j$ . Observe that $k(b)\neq u$ and $h(c)\neq u$ , for each $b\in B$ , $c\in C$ . Consider an ultrafilter $F_{0}$ over $\mathbf {D}_{i_0}$ (the lowest fiber in ${\mathbf D}$ ) and $F_{i}=p_{i_{0}i}[F_{0}]$ , for each $i_0\leq i$ and set $F=\displaystyle \bigcup _{i<u} F_{i}$ . Each $F_{i}$ is an ultrafilter over the algebra $\mathbf {D}_{i}$ (since homomorphisms are surjective). Consider the algebra $\mathbf {D}^{\prime }=\mathbf {D}\times \mathbf {B}_{2}$ , which shares with $\mathbf {D}$ the semilattice structure I and whose homomorphisms are denoted by $q_{ij}$ for $i\leq j$ . Observe that this algebra does not contain trivial fibers as u is the top element of I and $\mathbf {D}^{\prime }_{u}$ is the two-elements Boolean algebra with universe $\{\langle u,\top \rangle $ , $\langle u,\bot \rangle \}$ . This entails that $\mathbf {D}^{\prime }\in \mathsf {NBCA}$ . Define the map $k'\colon \mathbf {B}\to \mathbf {D}^{\prime }$ such that, for any $b\in B$ :

$$\begin{align*}k^{\prime}(b) = \begin{cases} \langle k(b),\top\rangle \text{, if } k(b)\in F, \\ \langle k(b),\bot\rangle, \text{ otherwise.} \end{cases} \end{align*}$$

Similarly, consider $h^{\prime }\colon \mathbf {C}\to \mathbf {D}^{\prime }$ defined by the same rule when applied to elements in C. We show that $k^{\prime } $ is an embedding. Clearly the map in injective, as k is. It is also clear that $k^{\prime }$ preserves the Boolean operations. Moreover, for $b\in B$ ,

$$ \begin{align*} k^{\prime}(J_{_2}b)=\langle k(J_{_2}b),\top\rangle &\iff \\ k(J_{_2}b)=J_{_2}k(b)\in F &\iff\\ k(b)\in F & \iff\\ J_{_2}k^{\prime}(b)=J_{_2}\langle(k(b),\top\rangle= \langle J_{_2}k(b),J_{_2}\top\rangle&=\langle k(J_{_2}b),\top\rangle, \end{align*} $$

where the second equivalence is justified because, for any $i\in I$ and $d_{i}\in D_{i}$ , $J_{_2}d_{i}\in p^{-1}_{0i}(d_{i})$ . This, together with an analogous argument applied to $h^{\prime }$ , show that $k^{\prime }, h^{\prime }$ are embeddings. In order to conclude the proof, recall that for $a\in A$ , $k\circ f(a)=h\circ g(a)$ , so the following equivalences hold:

$$ \begin{align*} k^{\prime}\circ f(a)=\langle k\circ f(a),\top\rangle \iff&\\ k\circ f(a)\in F\iff&\\ g^{\prime\prime}\circ g(a)=\langle g^{\prime}\circ g(a),\top\rangle=\langle k\circ f(a),\top\rangle. \end{align*} $$

This proves $k^{\prime }\circ f(a)=h^{\prime }\circ g(a)$ , as desired.