1 Introduction
The recent years have seen a renaissance of interests and studies around weak Kleene logics, logical formalisms that were considered, in the past, not particularly attractive in the panorama of threevalued logics, due to reputed “odd” behavior of the thirdvalue. The late (re)discovery of weak Kleene logic regards, almost exclusively, internal rather than external logics: the latter, in essence, consisting of linguistic expansions of the former. More precisely, here, for external Kleene logics we understand the external version of Bochvar logic (introduced by Bochvar himself [Reference Bochvar4]) and of Paraconsistent weak Kleene logic (introduced by Segerberg [Reference Segerberg35]).
The idea of considering the external connectives, thus enriching the (internal) logical vocabulary, is originally due to Russian logician D. Bochvar [Reference Bochvar4]. His aim was, from the one side, to adopt a nonclassical base to get rid of settheoretic and semantic paradoxes (by interpreting them to ) and, from the other, to preserve the expressiveness of classical logic. Although his attempt failed in reaching the former purpose (as it can be shown that paradoxes resurface [Reference Urquhart, Gabbay and Guenthner37]), the work of Bochvar has left us with a logic extremely rich in expressivity and whose potential has yet to be discovered and applied in its full capacity. Indeed, external weak Kleene logics have the advantage of limiting the infectious behavior of the third value—the feature making them apparently little attractive—which is confined to internal formulas only, and to recover all the consequences of classical logic in the purely external part of the language (this holding true for Bochvar external logic only). We believe that these features may turn out to be very useful in computer science and AI, providing new tools for modeling errors, concurrence, and debugging.
From a mathematical viewpoint, internal weak Kleene logics show quite a weak connection with respect to their algebraic counterparts, as they are examples of the socalled nonprotoalgebraic logics. On the other hand, a recent work [Reference Bonzio, Fano, Graziani and Pra Baldi6] has shown that Bochvar external logic is algebraizable with the quasivariety of Bochvar algebras (introduced in [Reference Finn and Grigolia17]) as its equivalent algebraic semantics. This observation gives a justified starting point for a deeper, intertwined, investigation of Bochvar external logic and Bochvar algebras, which is the main scope of the present work.
The flourishing trend of algebraic research around weak Kleene logics has strongly connected them with the algebraic theory of Płonka sums (see, e.g., [Reference Bonzio, Paoli and Pra Bldi9]). Recently, the tools offered by Płonka sums have fruitfully been extended to the structural analysis of residuated structures, establishing a natural connection with substructural logics (see [Reference GilFérez, Jipsen, Lodhia, Glück, Santocanale and Winter19, Reference Jenei22]). In line with this trend, we will further extend the application of the method. Not surprisingly, since Bochvar external logic is a linguistic expansion of Bochvar logic, we will show that the construction of the Płonka sum will play an important role in characterizing the structure of Bochvar algebras.
The paper is organized into five sections: in Section 2, we present Bochvar external logic as the logic induced by a single matrix, and we recall the axiomatization due to Finn and Grigolia. In Section 3, we first introduce the quasivariety of Bochvar algebras and prove some basic facts, including that any Bochvar algebra has an involutive bisemilattice reduct. We then proceed by describing the structure of Bochvar algebras: the main result is a representation theorem in terms of Płonka sums of Boolean algebras plus some additional operations. Section 4 is concerned with the study of the lattice of subquasivarieties of the quasivariety of Bochvar algebras, which is dually isomorphic to the lattice of extension of Bochvar external logic. We show that there are only three nontrivial quasivarieties of Bochvar algebras and we address the problem of (passive) structural completeness for each of them. Finally, in Section 5, we show that every quasivariety of Bochvar algebras has the amalgamation property. We conclude the paper with Appendix A, where we provide a new (quasi)equational basis for the quasivariety of Bochvar algebras. The proposed axiomatization significantly simplifies the traditional one introduced by Finn and Grigolia [Reference Finn and Grigolia17].
2 Bochvar external logic
Kleene’s threevalued 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 “twovalued” formulas—which he referred to as external formulas—beside the genuinely “threevalued” 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 threeelements algebra displayed in Figure 1.
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 socalled 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).
Definition 2.1. Bochvar external logic is the logic induced by the matrix ${\langle {\mathbf {WK}}^{\mathrm {e}},\{1\}\rangle }$ .
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 threevalued logics can be found in [Reference Ciucci and Dubois12]).
The following technicalities are needed to introduce a Hilbertstyle axiomatization of .
Definition 2.2. An occurrence of a variable x in a formula $\varphi $ is open if it does not fall under the scope of $J_{_k}$ , for every $k\in \{0,1,2\}$ . A variable x in $\varphi $ is covered if all of its occurrences are not open, namely if every occurrence of x in $\varphi $ falls under the scope of $J_{_k}$ , for some $k\in \{0,1,2\}$ .
The intuition behind the notion of external formulas is made precise by the following.
Definition 2.3. A formula $\varphi \in Fm$ is called external if all its variables are covered.
A Hilbertstyle 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(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 ) $ ;

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

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

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

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

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

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

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

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

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

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

(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 Hilbertstyle 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 Hilbertstyle 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
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.
Theorem 2.4 (Deduction Theorem).
if and only if .
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 classoperator 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 .
Definition 3.1. A Bochvar algebra ${\mathbf A}={\langle A,\vee ,\wedge , \neg , J_{_0},J_{_1},J_{_2}, 0,1\rangle }$ is an algebra of type ${\langle 2,2,1,1,1,1,0,0\rangle }$ satisfying the following identities and quasiidentities:

(1) $\varphi \vee \varphi \thickapprox \varphi $ ;

(2) $\varphi \lor \psi \thickapprox \psi \lor \varphi $ ;

(3) $(\varphi \lor \psi )\lor \delta \thickapprox \varphi \lor (\psi \lor \delta )$ ;

(4) $\varphi \land (\psi \lor \delta )\thickapprox (\varphi \land \psi )\lor (\varphi \land \delta )$ ;

(5) $\neg (\neg \varphi )\thickapprox \varphi $ ;

(6) $\neg 1\thickapprox 0$ ;

(7) $\neg ( \varphi \lor \psi )\thickapprox \neg \varphi \land \neg \psi $ ;

(8) $0\vee \varphi \thickapprox \varphi $ ;

(9) $J_{_2}J_{_k} \varphi \thickapprox J_{_k} \varphi $ , for every $k\in \{0,1,2\}$ ;

(10) $J_{_0}J_{_k} \varphi \thickapprox \neg J_{_k} \varphi $ , for every $k\in \{0,1,2\}$ ;

(11) $J_{_1}J_{_k} \varphi \thickapprox 0$ , for every $k\in \{0,1,2\}$ ;

(12) $J_{_k}(\neg \varphi )\thickapprox J_{_{2k}}\varphi $ , for every $k\in \{0,1,2\}$ ;

(13) $J_{_i}\varphi \thickapprox \neg (J_{_j}\varphi \vee J_{_k}\varphi )$ , for $i\neq j\neq k\neq i$ ;

(14) $J_{_k}\varphi \vee \neg J_{_k}\varphi \thickapprox 1$ , for every $k\in \{0,1,2\}$ ;

(15) $(J_{_i}\varphi \vee J_{_k}\varphi )\land J_{_i}\varphi \thickapprox J_{_i}\varphi $ , for $i,k\in \{0,1,2\}$ ;

(16) $ \varphi \lor J_{_k}\varphi \thickapprox \varphi $ , for $k\in \{1,2\}$ ;

(17) $J_{_0}(\varphi \lor \psi )\thickapprox J_{_0}\varphi \land J_{_0}\psi $ ;

(18) $J_{_2}(\varphi \lor \psi )\thickapprox ( 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) $J_{_0} \varphi \thickapprox J_{_0} \psi \;\&\; J_{_1} \varphi \thickapprox J_{_1} \psi \;\&\; J_{_2} \varphi \thickapprox J_{_2} \psi \;\Rightarrow \; \varphi \thickapprox \psi $ .
$\mathsf {BCA}$ 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.
Example 3.2. Let ${\mathbf A}$ be a (nontrivial) Boolean algebra. Setting the functions ${J_{_k}\colon A\to A}$ , with $k\in \{0,1,2\}$ as $J_{_2} = id$ , $J_{_1} = 0$ (the constant function onto $0$ ) and $J_{_0} = \neg $ , then ${\mathbf A} = {\langle A,\wedge ,\vee , \neg , 0,1, J_{_2}, J_{_1}, J_{_0}\rangle }$ is a Bochvar algebra.
For this reason, by $\mathbf {B}_{n}$ we will safely denote both the nelements Boolean algebra and its $\mathsf {BCA}$ expansion obtained according to Example 3.2. Since ${\mathbf {WK}}^{\mathrm {e}}$ generates $\mathsf {BCA}$ , a quasiequation holds in ${\mathbf {WK}}^{\mathrm {e}}$ if and only if it holds in every ${\mathbf A}\in \mathsf {BCA}$ .
The original quasiequational 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 quasiequational basis for $\mathsf {BCA}$ . This is accomplished in the next theorem.
Theorem 3.3. The following is a quasiequational basis for $\mathsf {BCA}$ .

(1) $\varphi \vee \varphi \thickapprox \varphi $ ;

(2) $\varphi \lor \psi \thickapprox \psi \lor \varphi $ ;

(3) $(\varphi \lor \psi )\lor \delta \thickapprox \varphi \lor (\psi \lor \delta )$ ;

(4) $\varphi \land (\psi \lor \delta )\thickapprox (\varphi \land \psi )\lor (\varphi \land \delta )$ ;

(5) $\neg (\neg \varphi )\thickapprox \varphi $ ;

(6) $\neg 1\thickapprox 0$ ;

(7) $\neg ( \varphi \lor \psi )\thickapprox \neg \varphi \land \neg \psi $ ;

(8) $0\vee \varphi \thickapprox \varphi $ ;

(9) $J_{_0}J_{_2} \varphi \thickapprox \neg J_{_2} \varphi $ ;

(10) $J_{_2}\varphi \thickapprox \neg (J_{_0}\varphi \vee J_{_1}\varphi )$ ;

(11) $J_{_2}\varphi \vee \neg J_{_2}\varphi \thickapprox 1$ ;

(12) $J_{_2}(\varphi \lor \psi )\thickapprox ( 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 ) $ ;

(13) $J_{_0} \varphi \thickapprox J_{_0} \psi \;\&\; J_{_2} \varphi \thickapprox J_{_2} \psi \;\Rightarrow \; \varphi \thickapprox \psi $ ,
where $J_{_0}\varphi \thickapprox J_{_2}\neg \varphi $ and $J_{_1}\varphi \thickapprox \neg (J_{_2}\varphi \lor J_{_0}\varphi )$ .
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.
Definition 3.4. An involutive bisemilattice is an algebra $\mathbf {B} = {\langle B,\land ,\lor ,\lnot ,0,1\rangle }$ of type $(2,2,1,0,0)$ satisfying:

I1. $\varphi \lor \varphi \thickapprox \varphi $ ;

I2. $\varphi \lor \psi \thickapprox \psi \lor \varphi $ ;

I3. $\varphi \lor (\psi \lor \delta )\thickapprox (\varphi \lor \psi )\lor \delta $ ;

I4. $\lnot \lnot \varphi \thickapprox \varphi $ ;

I5. $\varphi \land \psi \thickapprox \lnot (\lnot \varphi \lor \lnot \psi )$ ;

I6. $\varphi \land (\lnot \varphi \lor \psi )\thickapprox \varphi \land \psi $ ;

I7. $0\lor \varphi \thickapprox \varphi $ ;

I8. $1\thickapprox \lnot 0$ .
The class of involutive bisemilattices forms a variety, which we denote by $\mathsf {IBSL}$ . $\mathsf {IBSL}$ is the socalled 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, GilFérez, 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 singlefixpoint 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.
Proposition 3.5. Every Bochvar algebra has a $\mathsf {SIBSL}$ reduct.
Proof. It suffices to check that every valid $\mathsf {SIBSL}$ quasiequation is also valid in $\mathsf {BCA}$ . To see this, let $\chi $ be a quasiequation in the language of ${\mathbf {WK}}$ . We have that
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 (singlefixpoint) 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, GilFérez, 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 lowerbounded 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 nary operation g is defined as
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, GilFérez, 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.
Lemma 3.6. Let ${\mathbf A}$ be a (nontrivial) Bochvar algebra and ${\mathcal {P}}_{\unicode{x0142}}({\mathbf A}_{i})_{i\in I}$ be the Płonka decomposition of the $\mathsf {IBSL}$ reduct of ${\mathbf A}$ , having $i_{0}$ as the least element in I. Then:

(1) $J_{_1}^{{\mathbf A}_{i_0}}$ is the constant map onto $0$ ;

(2) $J_{_2}^{{\mathbf A}_{i_0}} = id$ ;

(3) $J_{_0}^{{\mathbf A}_{i_0}} a = \neg a$ , for any $a\in A_{i_{0}}$ .
Proof. It suffices to notice that (1) holds if and only if
and this quasiequation clearly holds in ${\mathbf {WK}}^{\mathrm {e}}$ . The same applies to (2), (3) with respect to the quasiequations $\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 $J_{_1}$ .
Lemma 3.7. The following hold for every ${\mathbf A}\in \mathsf {BCA}$ (with ${\mathcal {P}}_{\unicode{x0142}}({\mathbf A}_{i})_{i\in I}$ the Płonka decomposition of its $\mathsf {IBSL}$ reduct).

(1) if $a,b\in A_{i}$ , for some $i\in I$ , then $J_{_1}a=J_{_1}b$ ;

(2) if $a\in A_{i_{0}}$ then $J_{_1}a=0$ ;

(3) if $a=\neg a$ then $J_{_1}a=1$ ;

(4) $p_{i_{0}i}(J_{_1}a)=a\land \neg a$ , for every $a\in A_{i}$ and $i\in I$ .
Proof. (1). For every $a,b\in A_{i}$ , $a\land (a\lor b)=a$ and $b\land (b\lor a)=a$ . Since
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
so $p_{i_{0}i}(J_{_1}a)=J_{_1}a\land (J_{_1}a\lor a)=a\land \neg a$ .
Remark 3.8. It follows from Lemma 3.7 that, for every $a,b\in A_{i}$ (for some $i\in I$ ), i.e., $a,b$ are elements in the same fiber of the Płonka sum, $J_{_1} a = J_{_1} b $ , thus, in particular, $J_{_1}(a\vee b) = J_{_1} a \vee J_{_1} b = J_{_1} a \wedge J_{_1} b = J_{_1}(a\wedge b)$ .
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.
Lemma 3.9. Let ${\mathbf A}$ be a Bochvar algebra with ${\mathcal {P}}_{\unicode{x0142}}({\mathbf A}_{i})_{i\in I}$ the Płonka decomposition of its $\mathsf {IBSL}$ reduct. Then:

(1) ${\mathcal {P}}_{\unicode{x0142}}({\mathbf A}_{i})_{i\in I}$ has surjective homomorphisms;

(2) for every $i\neq i_{0}$ , $p_{i_{0}i}$ is not injective;

(3) for every $a\in A$ and every $i\in I$ (with $a\in A_i$ ), $J_{_2} a\in p_{i_0 i}^{1}(a)$ and $J_{_0} a\in p_{i_0 i}^{1}(\neg a)$ ;

(4) for every $a,b\in A_i$ , $J_{_2} (a\vee b ) = J_{_2} a\vee J_{_2} b$ .
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 quasiequation
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 quasiequation (13), $0 = 0_{j}$ , a contradiction.
Remark 3.10. It follows from Lemma 3.9 that, for any $i\in I$ , $ p_{i_{0}i}\circ J_{_2}^{{{\mathbf A}_{i}}} = id_{{\mathbf A}_i} $ , namely that $J_{_2}$ (restricted on ${\mathbf A}_i$ ) is the right inverse of the surjective homomorphism $p_{i_{0}i}$ .
Remark 3.11. Note that, in general, it does not hold that $J_{_2} (\varphi \vee \psi )\thickapprox J_{_2} \varphi \vee J_{_2} \psi $ (the identity is falsified in ${\mathbf {WK}}^{\mathrm {e}}$ ).
Recall that, for any nontrivial 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.
Proposition 3.12. Let ${\mathbf A}$ be a Bochvar algebra with ${\mathcal {P}}_{\unicode{x0142}}({\mathbf A}_{i})_{i\in I}$ the Płonka decomposition of its $\mathsf {IBSL}$ reduct. Then, for every $i\in I$ , $J_{_2}\colon {\mathbf A}_i \to \mathbf {[0,\top ]}$ is an isomorphism onto the interval Boolean algebra $\mathbf {[0,\top ]} = {\langle [0,\top ], \wedge ,\vee , ^{\ast }, 0 , \top \rangle }$ , where $\top = J_{_2} 1_i$ , for every $i\neq i_{0}$ .
Proof. ${\mathbf {WK}}^{\mathrm {e}}\models J_{_2}(\varphi \land \psi )\thickapprox J_{_2}\varphi \land J_{_2}\psi $ , thus $J_{_2}$ preserves the $\land $ operation. By Lemma 3.9(4) it also preserves $\lor $ when the arguments belong to the same fiber. This implies that $J_{_2}({\mathbf A}_i)$ is a lattice. To see that is bounded, recall that $J_{_2} 0_i = 0$ (with $0_i$ the bottom element of ${\mathbf A}_i$ ); moreover, let $b\in J_{_2}(A_{i})$ , thus ${b = J_{_2} a}$ , for some $a\in A_i$ , then $J_{_2} a\vee J_{_2} 1_i = J_{_2} (a\vee 1_i) = J_{_2} 1_i$ , i.e., $J_{_2} a\leq J_{_2} 1_i$ thus $\top = J_{_2} 1_i$ is the top element of the lattice $J_{_2}({\mathbf A}_i)$ . Finally, observe that, for any $a\in A_i$ , $J_{_2} \neg a = J_{_0} a = \neg J_{_2} a\wedge \neg J_{_1} a = \neg J_{_2} a\wedge (J_{_2} a \vee J_{_0} a) = \neg J_{_2} a \wedge (J_{_2} a\vee J_{_2}\neg a) = \neg J_{_2} a \wedge (J_{_2} (a\vee \neg a)) =\neg J_{_2} a \wedge J_{_2} 1_i = (J_{_2} a)^{\ast } $ , i.e., $J_{_2}$ preserves also the complementation of the interval algebra $\mathbf [0,\top ]$ . We have so shown that $J_{_2} $ is a boolean homomorphism. Moreover, $J_{_2}$ is injective as $p_{i_{0}i}$ is its leftinverse (see Remark 3.10). To see that $J_{_2}$ is also surjective (onto $[0, \top ]$ ), let $a \in [0, \top ]$ , i.e., $a\leq J_{_2} 1_i$ . By Lemma 3.6, $J_{_2} a = a\leq J_{_2} 1_i$ hence $a = J_{_2} a\wedge J_{_2} 1_i = J_{_2} (a\wedge 1_i) = J_{_2} (p_{i_{0}i}(a)\wedge 1_i) = J_{_2} (p_{i_{0}i}(a))$ . This concludes the proof, because $p_{i_{0}i}(a)\in A_{i}$ .
Remark 3.13. Since $p_{i_{0}i}$ is a surjective (but not injective) homomorphism, for any $i\in I$ (and $i\neq i_0$ ), then ${\mathbf A}_{i_{0}}/Ker(p_{i_{0}i})\cong {\mathbf A}_i$ (with $Ker(p_{i_{0}i}\neq \Delta ^{{\mathbf A}_{i_0}}$ for $i\neq i_{0}$ ), via the isomorphism f mapping $[x]_{Ker(p_{i_{0}i})}\mapsto p_{i_{0}i}(x)$ . The proof of Proposition 3.12 shows that $J_{_2}$ is in fact the inverse of f.
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.
Lemma 3.14. Let ${\mathbf A}$ be a Bochvar algebra with ${\mathcal {P}}_{\unicode{x0142}}({\mathbf A}_{i})_{i\in I}$ the Płonka decomposition of its $\mathsf {IBSL}$ reduct. Then:

(1) If $i < j$ , then $J_{_2} 1_j < J_{_2} 1_i$ . In particular, $J_{_2}({\mathbf A}_j) = [0,J_{_2} 1_j]\subset [0,J_{_2} 1_i] = J_{_2}({\mathbf A}_i)$ .

(2) $J_{_2}(p_{ij}(a))\leq J_{_2} a$ , for every $i\leq j$ and $a\in A_{i}$ .
Proof. (1). If $1_{i}<1_{j}$ then $1_{i}\land 1_{j}=1_{j}\neq 1_{i}$ . The following two quasiequations hold in ${\mathbf {WK}}^{\mathrm {e}}$ :
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.
Theorem 3.15 (Płonka sum decomposition).
Let ${\mathbf A}$ be a Bochvar algebra with ${\mathcal {P}}_{\unicode{x0142}}({\mathbf A}_i)_{i\in I}$ the Płonka sum representation of its $\mathsf {IBSL}$ reduct. Then:

(1) All the homomorphisms $\{p_{ij}\}_{i\leq j}$ are surjective and $p_{i_{0}i}$ is not injective for every $i\neq i_{0}$ .

(2) For every $i\in I$ , there exists an element $a_{i}\in A_{i_{0}}$ such that the restriction $p_{i_{0}i} $ on $\mathbf {[0,a_{i}]}$ is an isomorphism (with inverse $J_{_2}$ ) onto ${\mathbf A}_{i}$ , with $a_{i}\neq a_{j}$ for every $i\neq j$ ; in particular, if $i<j$ then $a_{j}<a_{i}$ .
Moreover, the decomposition is unique up to isomorphism.
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 singlefixpoint 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, GilFérez, 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, GilFérez, 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.
Remark 3.16. Observe that, as a consequence of Theorem 3.15, the Płonka sum decomposition of a Bochvar algebra ${\mathbf A}$ admits no injective homomorphism (excluding the identical homomorphisms $p_{ii}$ ). To see this, suppose $p_{ij}$ is injective, for some $i<j$ . Then $p_{ij}$ is an isomorphism, as it is also a surjective map. Observe that $a_{j}=J_{_2}1_{j}<a_{i}=J_{_2}1_{i}$ and $p_{i_{0}i}\colon \mathbf {[0,a_{i}]}\to {\mathbf A}_{i}$ is also an isomorphism. Therefore, $p_{i_{0}i}(a_{j})\neq 1_{i}=p_{i_{0}i}(a_{i})$ and, by the injectivity of $p_{ij}$ , $p_{ij}\circ p_{i_{0}i}(a_{j})=p_{i_{0}j}(a_{j})\neq 1_{j}$ , a contradiction.
We now show that the conditions displayed in the above theorem are also sufficient to equip any $\mathsf {SIBSL}$ with a $\mathsf {BCA}$ structure.
Theorem 3.17. Let ${\mathbf A} = {\langle A,\wedge ,\vee ,\neg , 0,1\rangle }$ be an involutive bisemilattice whose Płonka sum representation is such that:

(1) all homomorphisms are surjective and $p_{i_{0}i}$ is not injective for every $i_{0}\neq i\in I$ ;

(2) for each $i\in I$ there exists an element $a_{i}\in A_{i_{0}}$ such that $p_{i_{0} i}\colon \mathbf {[0},\mathbf {a}_{i}]\to {\mathbf A}_i$ is an isomorphism, with $a_{i}\neq a_{j}$ for $i\neq j$ and, in particular, $a_{j}<a_{i}$ for each $i<j$ .
Define, for every $a\in A_{i}$ and $i\in I$ :

• $J_{_2}(a)=p_{i_{0}i}^{1}(a)\in [0,a_{i}]$ ;

• ;

• .
Then $\mathbf {B} = {\langle A,\wedge ,\vee ,\neg , 0,1, J_{_2}, J_{_1}, J_{_0}\rangle }$ is a Bochvar algebra.
Proof. It is immediate to check that assumption (1) implies that ${\mathbf A}\in \mathsf {SIBSL}$ . Since $p_{i_{0}i}\colon [0,\mathbf {a}_{i}]\to {\mathbf A}_{i}$ is an isomorphism (with inverse $p_{i_{0}i}^{1}$ ) the maps $J_{_2}$ , $J_{_1}$ and $J_{_0}$ are well defined and naturally extend to the whole algebra ${\mathbf A}$ . It can be mechanically checked that $\mathbf {B}$ satisfies all the quasiequations in Definition 3.1.
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.
Corollary 3.18. Let ${\mathbf A}\in \mathsf {SIBSL}$ be with surjective and noninjective homomorphisms. The following are equivalent:

(1) ${\mathbf A}$ is the reduct of a Bochvar algebra;

(2) for each $i\in I $ , $1/\operatorname {\mathrm {Ker}}_{p_{i_{0}i}}$ is a principal filter, with generator $a_{i}\in A_{i_{0}}$ . Moreover, if $i\neq j$ then $a_{i}\neq a_{j}$ and $a_{j}<a_{i}$ for each $i<j$ ;

(3) for each $i\in I$ there exists an element $a_{i}\in A_{i_{0}}$ such that $p_{i_{0} i}\colon \mathbf {[0},\mathbf {a}_{i}]\to {\mathbf A}_i$ is an isomorphism. Moreover, if $i\neq j$ then $a_{i}\neq a_{j}$ and $a_{j}<a_{i}$ for each $i<j$ .
Proof. We just show $(2)\Leftrightarrow (3)$ . Let $1/\operatorname {\mathrm {Ker}}_{p_{i_{0}i}}$ be the filter generated by $a_{i}$ , for some $a_i \in A_{i_{0}}$ . It is routine to check that $p_{i_{0}i}\colon \mathbf {[0},\mathbf {a}_{i}]\to {\mathbf A}_i$ is an isomorphism. Conversely, assume that there is an element $a_i\in A_{i_0}$ such that $p_{i_{0}i}\colon \mathbf {[0},\mathbf {a}_{i}]\to {\mathbf A}_i$ is an isomorphism. Suppose, by contradiction, that $1/\operatorname {\mathrm {Ker}}_{p_{i_{0}i}}$ is not principal, i.e., there is an element $b\in A_{i_{0}}$ such that $a_i\nleq b$ and $p_{i_{0}i}(b) = 1_i$ . Then $c = b\wedge a_i$ is an element in $[0, a_i]$ such that $p_{i_{0}i}(c) = 1_i $ , in contradiction with the fact that $p_{i_{0}i}$ is an isomorphism.
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.
Example 3.19. Let $\mathcal {P}(\mathbb {Z})$ be the power set Boolean algebra over the integers. This algebra is uncountable and atomic, with $\mathbb {Z}$ as top element. Consider now the nonprincipal ideal I containing all the finite subsets of $\mathbb {Z}$ . Observe that $\mathcal {P}(\mathbb {Z})/I$ is an atomless Boolean algebra. Let ${\mathbf A}$ be the Płonka sum built over the two fibers $\mathcal {P}(\mathbb {Z}), \mathcal {P}(\mathbb {Z})/I$ and with (the unique nontrivial) homomorphism the canonical map $p\colon \mathcal {P}(\mathbb {Z})\to \mathcal {P}(\mathbb {Z})/I$ . Clearly ${\mathbf A}$ satisfies condition (1) of Theorem 3.17 (the homomorphism p is both surjective and noninjective hence ${\mathbf A}\in \mathsf {SIBSL}$ ). However the filter $\mathbb {Z}/\operatorname {\mathrm {Ker}}_{p}$ (corresponding to the ideal I) is not principal, as $I $ is not. In the light of condition (2) of Corollary 3.18, ${\mathbf A}$ is a $\mathsf {SIBSL}$ which is not the reduct of any 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 threeelements 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 $\mathsf {K}^{\prime }$ :
where $\mathbb {V}(\mathsf {K})=HSP(\mathsf {K})$ is the variety generated by $\mathsf {K}$ . Recall that a quasiidentity $ \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 quasiidentity is passive if it is equivalent to a quasiidentity 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).
Definition 4.2 [Reference Wroński38].
A quasivariety $\mathsf {K}$ is PSC if every passive quasiidentity over $\mathsf {K}$ is valid in $\mathsf {K}$ .
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 nontrivial 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.
Theorem 4.3 [Reference Moraschini, Raftery and Wannenburg25, theorem 7.11].
Let $\mathsf {K}$ be a quasivariety of finite type and ${\mathbf A}\in \mathsf {K}$ a finite $0$ generated algebra. Then $Ret(\mathsf {K},{\mathbf A})$ is a maximal PSC subquasivariety of $\mathsf {K}$ .Footnote ^{5}
The following summarizes the properties of the quasivariety $\mathsf {BCA}$ with respect to the above introduced notions.
Proposition 4.4. The following hold:

(i) $\mathsf {BCA}$ is not PSC;

(ii) $\mathbf {B}_{2}$ is the $0$ generated algebra in $\mathsf {BCA}$ ;

(iii) $Ret(\mathsf {BCA},\mathbf {B}_{2})$ is a maximal PSC subquasivariety of $\mathsf {BCA}$ .
Proof. $\mathrm{(i)}$ . The quasiidentity
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 quasiidentity
Since $(NF)$ is valid in the quasivariety $Ret(\mathsf {BCA},\mathbf {B}_{2})$ , it demands that the underlying involutive bisemilattice of any of its nontrivial members lacks trivial fibers. A stronger fact is established in the following lemma.
Lemma 4.5. Let ${\mathbf A}\in \mathsf {BCA}$ be nontrivial. Then ${\mathbf A}\in Ret(\mathsf {BCA},\mathbf {B}_{2})$ if and only if ${\mathbf A}\vDash (NF)$ .
Proof. $(\Rightarrow )$ . By Proposition 4.4, $Ret(\mathsf {BCA}, \mathbf {B}_{2})$ is a (maximal) PSC quasivariety and $(NF)$ is a passive quasiidentity, 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
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 ${\mathbf A}\in Ret(\mathsf {BCA},\mathbf {B}_{2})$ .
Corollary 4.6. $Ret(\mathsf {BCA},\mathbf {B}_{2})$ is the quasivariety axiomatized by adding $(NF)$ to the quasiequational theory of $\mathsf {BCA}$ . Moreover, a Bochvar algebra belongs to $Ret(\mathsf {BCA},\mathbf {B}_{2})$ if and only if its Płonka sum decomposition lacks trivial fibers.
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 fourelement Boolean algebra $\mathbf {B}_{4}$ , the twoelement one $\mathbf {B}_{2}$ , the twoelement 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.
Theorem 4.8. The quasivariety $\mathsf {NBCA} $ is generated by $\mathbf {B}_{4}\oplus \mathbf {B}_{2}$ .
Proof. We show that $\mathsf {NBCA}=ISP(\mathbf {B}_{4}\oplus \mathbf {B}_{2})$ . The right to left inclusion is obvious, as subalgebras of direct products of an involutive bisemilattice without trivial fibers preserve the property of lacking trivial fibers.
For the converse, the proof is an adaption of [Reference Paoli and Pra Baldi26, theorem 7], and we only sketch its main ingredients (leaving the details to the reader). Preliminarily recall that for quasivarieties $\mathsf {K},\mathsf {K}^{\prime }$ , $\mathsf {K}\subseteq \mathsf {K}^{\prime }$ if and only if every finitely generated member of $\mathsf {K}$ belongs to $\mathsf {K}^{\prime }$ . Moreover, for algebras ${\mathbf A},\mathbf {B}$ , ${\mathbf A}\in ISP(\mathbf {B})$ if and only if that there exists a family $H\subseteq Hom({\mathbf A},\mathbf {B})$ such that $\displaystyle \bigcap _{h\in H}Ker(h)=\Delta ^{{\mathbf A}}$ . It is possible to show that, for a finitely generated ${\mathbf A}\in \mathsf {NBCA}$ , there exists a family of homomorphisms $H\subseteq Hom({\mathbf A},\mathbf {B}_{4}\oplus \mathbf {B}_{2})$ such that $\displaystyle \bigcap _{h\in H}Ker(h)=\Delta ^{{\mathbf A}}$ . Indeed, $\mathbf {B}_{4}\oplus \mathbf {B}_{2}\cong {\mathbf {WK}}^{\mathrm {e}}\times \mathbf {B}_{2}$ and ${\mathbf {WK}}^{\mathrm {e}}$ generates $\mathsf {BCA}$ , so there exists $H\subseteq Hom({\mathbf A},\mathbf {WK})$ such that $\displaystyle \bigcap _{h\in H}Ker(h)=\Delta ^{{\mathbf A}}$ . Moreover, by Lemma 4.5, $\mathbf {B}_{2}$ is a retract of ${\mathbf A}$ , so there exists a retraction $r\colon {\mathbf A}\to \mathbf {B}_{2}$ . Now, the family $H\times \{r\}$ is a family of homomorphisms from ${\mathbf A}$ to $\mathbf {B}_{4}\oplus \mathbf {B}_{2}$ , defined for each $h\in H$ and $a\in A$ by $a\mapsto \langle h(a),g(a)\rangle $ . It is easy to check that $\displaystyle \bigcap _{h\in H}Ker \langle h,g\rangle =\Delta ^{{\mathbf A}}$ , so ${\mathbf A}\in ISP(\mathbf {B}_{4}\oplus \mathbf {B}_{2})$ , as desired.
Upon noticing that every nontrivial $\mathsf {NBCA}$ 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 twoelement 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 quasiequational 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 nontrivial subquasivarieties of $\mathsf {BCA}$ , proving that it consists of the following threeelements chain.
Theorem 4.9. The only nontrivial subquasivarieties of $\mathsf {BCA}$ are $\mathsf {NBCA}$ and $\mathsf {JBA}$ . They form a three element chain $\mathsf {JBA}\subset \mathsf {NBCA}\subset \mathsf {BCA}$ .
Proof. We already proved $\mathsf {JBA}\subset \mathsf {NBCA}\subset \mathsf {BCA}$ . We only have to show that these are the only nontrivial ones. Suppose that $\mathsf {K}\subseteq \mathsf {BCA}$ and $\mathsf {K}\nsubseteq \mathsf {NBCA}$ . Therefore there exists ${\mathbf A}\in \mathsf {K}$ and ${\mathbf A}\notin \mathsf {NBCA}$ . This entails ${\mathbf A}\in \mathsf {BCA}$ and ${\mathbf A}$ has a unique trivial fiber with universe $\{a\}$ . Clearly $g\colon {\mathbf {WK}}^{\mathrm {e}}\to {\mathbf A}$ mapping $1^{{\mathbf {WK}}^{\mathrm {e}}}\to 1^{{\mathbf A}}, 0^{{\mathbf {WK}}^{\mathrm {e}}}\to 0^{{\mathbf A}}, 1/2\to a$ is an embedding. Therefore ${\mathbf {WK}}^{\mathrm {e}}\in S({\mathbf A})$ , whence ${\mathbf {WK}}^{\mathrm {e}}\in \mathsf {K}$ . Since ${\mathbf {WK}}^{\mathrm {e}}$ generates $\mathsf {BCA}$ , $\mathsf {K}=\mathsf {BCA}$ .
Suppose now $\mathsf {K}\subseteq \mathsf {NBCA}$ and $\mathsf {K}\nsubseteq \mathsf {JBA}$ and let ${\mathbf A}\in \mathsf {K},{\mathbf A}\notin \mathsf {JBA}$ . This entails that the Płonka sum decomposition of ${\mathbf A}$ has at least two fibers ${\mathbf A}_{i_{0}}, {\mathbf A}_{i} (i_{0}<i)$ and no fiber is trivial. Moreover, by Lemma 3.9, ${\mathbf A}_{i_{0}}$ has cardinality $\geq 4$ (for otherwise ${\mathbf A}_{i}$ would be trivial). Let $h\colon \mathbf {B}_{4}\oplus \mathbf {B}_{2}\to {\mathbf A}$ mapping $1^{\mathbf {B}_{4}\oplus \mathbf {B}_{2}}\to 1^{{\mathbf A}}, 0^{\mathbf {B}_{4}\oplus \mathbf {B}_{2}}\to 0^{{\mathbf A}}$ , $1^{\mathbf {B}_{2}}\to 1_{i}, 0^{\mathbf {B}_{2}}\to 0_{i}$ , $a\to J_{_2}(1_{i}), \neg a\to \neg J_{_2}1_{i}$ . Clearly h is an embedding, so $\mathbf {B}_{4}\oplus \mathbf {B}_{2}\in S({\mathbf A})$ , which entails $\mathsf {K}=\mathsf {NBCA}$ .
Corollary 4.10. The quasivariety $\mathsf {NBCA}$ is structurally complete.
Proof. The only proper subquasivariety of