On Morgado and Sette’s Implicative Hyperlattices as Models of da Costa Logic Cω

1. Introduction

Hyperstructures are algebraic systems characterized by at least one multioperation, also known as a hyperoperation, which produces, for a given input, a set of possible outcomes rather than a single one. The concept of hyperstructures, also referred to as multialgebras, was introduced by Frédéric Marty in 1934 during the 8th Congress of Scandinavian Mathematicians [1]. Marty’s pioneering work laid the foundation for the development of hypergroup theory and the broader study of hyperstructures. In the context of non-classical logics, hyperstructures play a pivotal role by providing a flexible semantical framework which intends to expand the horizons of Abstract Algebraic Logic (AAL). Indeed, hyperalgebras (and non-deterministic semantics in general) enable the representation of logical systems that cannot be adequately captured by traditional algebraic methods, thereby facilitating the study and application of a broader range of logical systems.

In the general context of ordered structures, the concept of hyperlattices was first introduced by Benado in 1953 [2]. Since then, alternative axiomatizations have been proposed, such as those by D. J. Hansen [3], who aimed to avoid the partial associativity property present in Benado’s formulation, and by other researchers seeking to refine and generalize the original concept. Notably, José Morgado, in his book “Introdução à Teoria dos Reticulados” [4], provides a definition of hyperlattice that, to the best of our knowledge, is entirely original.1 He introduced the term “reticuloide” (reticuloid) to denote hyperlattices, and introduced the concepts of “supremoide” and “infimoide” as the corresponding hyperoperations for supremum and infimum. His two equivalent definitions of hyperlattices are more intuitive than Benado’s, as they resemble the usual lattice definitions while generalizing supremum and infimum properties in quasi-ordered structures. Morgado’s formulation allows for multiple suprema and infima, leading to a natural extension of classical lattice structures.

José Morgado (1921–2003) was a distinguished Portuguese mathematician whose scientific contributions significantly advanced the field of lattice theory. His work encompassed books and numerous papers, with a focus on the structure and properties of lattices. In [4], Morgado offered a comprehensive introduction to lattice theory, presenting original definitions and concepts that have influenced subsequent research. His innovative approach to hyperlattices, particularly through the introduction of "reticuloides," has been instrumental in extending classical lattice structures to more generalized settings. Morgado’s work continues to be a valuable reference for researchers in the field.

The Brazilian mathematician Antonio Antunes Mario Sette (1939-1999) also made a significant contribution to the study of hyperlattices (and also to the field of non-classical logics). Motivated by the original notion of $C_n$-algebras introduced by Newton da Costa in [6] for his paraconsistent calculi $C_n$ (for $1\le n<\omega$), Sette and da Costa propose in [7] the $C_{\omega }$-algebras for the limiting paraconsistent logic system $C_{\omega }$. In 1971, Sette introduced in his master’s dissertation under da Costa’s supervision [8] the concept of $C_{\omega }$-hyperlattices, proving that they are in correspondence with $C_{\omega }$-algebras. His approach built upon the hyperlattice definition given by José Morgado. Sette, working under da Costa’s guidance, extended Morgado’s framework to introduce implicative hyperlattices, a natural generalization of implicative lattices within Morgado’s hyperlattice context. This innovation proved to be particularly relevant to the semantics of logic $C_{\omega }$, as we shall see.

In this paper, we bring to the broader research community the concepts of m-hyperlattices and SIHLs (Sette Implicative Hyperlattices), which were previously available only in Portuguese. We establish fundamental properties of these structures, which in turn allow us to prove the soundness and completeness of $C_{\omega }$ with respect to SHC_ω (Sette hyperalgebras for $C_{\omega }$) semantics. Furthermore, we introduce a natural subclass of SIHLs defined by swap structures, which also provide an adequate semantics for $C_{\omega }$.

2. Morgado Reticuloids

There are several definitions of hyperalgebra in the literature, considering that each hyperalgebra application in a specific area of Mathematics (mainly Algebra and Logic) requires a particular adaptation. Here, we adapt the notion of hyperalgebra used in [9]; the identity theory here presented is close to the exposed in [10].

Definition 1.

A hyperalgebraic signature is a sequence of pairwise disjoint sets

$$\Sigma ={\left({\Sigma }_n\right)}_{n\in \mathbb{N}},$$

where ${\Sigma }_n=S_n\bigsqcup M_n$, which $S_n$ is the set of strict hyperoperation symbols and $M_n$ is the set of hyperoperation symbols. In particular, ${\Sigma }_0=S_0\bigsqcup M_0$, $F_0$ is the set of symbols for constants and $M_0$ is the set of symbols for hyperconstants. A hyperalgebraic signature can also be denoted by

$$\Sigma =({\left(S_n\right)}_{n\ge 0},{\left(M_n\right)}_{n\ge 0}).$$

Definition 2.

Let A be any set, and $P^{*}\left(A\right):=\mathcal{P}\left(A\right)\setminus \{\varnothing \}$.

1. 

A hyperoperation of arity $n\in \mathbb{N}$ over a set A is a function $\#:A^n\to {\mathcal{P}}^{*}\left(A\right)$.

2. 

A hyperoperation # of arity $n\in \mathbb{N}$ over a set A is strict whenever it factors through the singleton function $s_A:A\to {\mathcal{P}}^{*}\left(A\right)$, $a\mapsto \left\{a\right\}$. Thus, it can be naturally identified with an ordinary n-ary operation $\#:A^n\to A$.

A 0-ary hyperoperation (respectively strict hyperoperation) on A can be identified with a non-empty subset of A (respectively a singleton subset of A).

Definition 3.

A  hyperalgebra  over a signature $\Sigma =({\left(S_n\right)}_{n\ge 0},{\left(M_n\right)}_{n\ge 0})$, is a set A endowed with a family of n-ary hyperoperations

$${\sigma }_n^A:A^n\to {\mathcal{P}}^{*}\left(A\right),{\sigma }_n\in S_n\bigsqcup M_n,n\in \mathbb{N},$$

such that: if ${\sigma }_n\in S_n$, then ${\sigma }_n^A:A^n\to {\mathcal{P}}^{*}\left(A\right)$ is a strict n-ary hyperoperation.

Remark 1.

1. 

Every algebraic signature $\Sigma ={\left(F_n\right)}_{n\in \mathbb{N}}$ is a hyperalgebraic signature where $M_n=\varnothing$, for every $n\in \mathbb{N}$. Each algebra

$$(A,{({(A^n\stackrel{f^A}{\to }A)}_{f\in F_n})}_{n\in \mathbb{N}})$$

over the algebraic signature Σ can be naturally identified with a hyperalgebra

$$(A,{({(A^n\stackrel{f^A}{\to }A\stackrel{s_A}{\to }{\mathcal{P}}^{*}(A))}_{f\in F_n})}_{n\in \mathbb{N}})$$

over the same signature.

2. 

Every hyperalgebraic signature $\Sigma =({(S_n)}_{n\in \mathbb{N}},{(M_n)}_{n\in \mathbb{N}})$ induces naturally a first-order language

$$L(\Sigma )=({(F_n)}_{n\in \mathbb{N}},{(R_{n+1})}_{n\in \mathbb{N}})$$

where $F_n:=S_n$ is the set of n-ary operation symbols and $R_{n+1}:=M_n$ is the set of (n+1)-ary relation symbols. In this way, hyper-algebras

$$(A,{({(A^n\stackrel{{\sigma }^A}{\to }{\mathcal{P}}^{*}(A))}_{\sigma \in S_n\bigsqcup M_n})}_{n\in \mathbb{N}})$$

over a hyperalgebraic signature $\Sigma ={(S_n\bigsqcup M_n)}_{n\in \mathbb{N}}$ can be naturally identified with the first-order structures over the language $L(\Sigma )$ that satisfies the $L(\Sigma )$-sentences:

$$\forall x_0\cdots \forall x_{n-1}\exists x_n({\sigma }_n(x_0,\dots ,x_{n-1},x_n)),for each{\sigma }_n\in R_{n+1},n\in \mathbb{N}.$$

Definition 4 (Prosets).

A preordered set (Proset) is a pair $\mathsf{P}=\langle P,⪯\rangle$ such that P is a non-empty set and ⪯ is a reflexive and transitive relation on P (i.e., a preorder). That is: $x⪯x$, for every $x\in P$; and $x⪯y$, $y⪯z$ implies $x⪯z$, for every $x,y,z\in P$.

If $x⪯y$ and $y⪯x$ we say that x and y are similar, and we write $x\equiv y$. Given $B,C\subseteq P$, $B⪯C$ means that $x⪯y$ for every $x\in B$ and every $y\in C$, and so $x⪯B$ denotes that $x⪯y$ for every $y\in B$, and $B⪯x$ denotes that $y⪯x$ for every $y\in B$.

Observe that $\varnothing ⪯B$ and $B⪯\varnothing$ for every $B\subseteq P$. Analogously, $x⪯\varnothing$ and $\varnothing ⪯x$ for every $x\in P$.

Definition 5.

Let $\mathsf{P}$ be a proset, and let $B\subseteq P$.

1. 

The set of minima of B is $\mathsf{Min}\left(B\right)=\{x\in B:x⪯B\}$, and the set of maxima of B is $\mathsf{Max}\left(B\right)=\{x\in B:B⪯x\}$.

2. 

The set of upper bounds of B is $Ub\left(B\right)=\{z\in P:B⪯z\}$. The set of lower bounds of B is $Lb\left(B\right)=\{z\in P:z⪯B\}$.

Observe that $\mathsf{Min}(\varnothing )=\mathsf{Max}(\varnothing )=\varnothing$, and $Ub(\varnothing )=Lb(\varnothing )=P$.

Definition 6 (Morgado hyperlattices, [4, Ch. II, §2, p. 122]).

Let $\mathsf{P}$ be a proset, and let $x,y\in P$.

1. 

The Morgado hypersupremum (or supremoid) of x and y is the set $x⋎y=\mathsf{Min}\left(Ub\right(\{x,y\}\left)\right)$.

2. 

The Morgado hyperinfimum (or infimoid) of x and y is the set $x⋏y=\mathsf{Max}\left(Lb\right(\{x,y\}\left)\right)$.

3. 

$\mathsf{P}$ is said to be a Morgado hyperlattice (or an m-hyperlattice) if $x⋎y$ and $x⋏y$ are nonempty sets for every $x,y\in P$.

Remark 2.

As observed in [4, Ch. II, §2, pp. 129–132], m-hyperlattices can be alternatively defined as hyperstructures $\mathsf{L}=\langle L,⋏,⋎\rangle$ satisfying the following properties:

R1 - 

$a⋎b=b⋎a$

R2 - 

if $x\in a⋎b$ and $y\in b⋎c$ then $x⋎c=a⋎y$

R3 - 

if $x\in a⋎b$ then $a\in a⋏x$

R4 - 

$a⋏b=b⋏a$

R5 - 

if $x\in a⋏b$ and $y\in b⋏c$ then $x⋏c=a⋏y$

R6 - 

if $x\in a⋏b$ then $a\in a⋎x$.

In this case, the preorder ⪯ is defined as follows: $x⪯y$ iff $x\in x⋏y$ (iff $y\in x⋎y$).

It is quite straightforward to deduce Axioms R1-R6 from Definition 6. But the converse is not so trivial. For instance, let us extract some basic properties from Axioms R1-R6 (we leave to the reader to check the main details of the equivalence between Axioms R1-R6 and Definition 6):

Lemma 1.

Let $\mathsf{L}=\langle L,⋏,⋎\rangle$ be a hyperalgebra satisfying Axioms R1-R6, and let $x,y\in L$. Then:

1. 

Let ⪯ be the relation defined as follows: $x⪯y$ iff $x\in x⋏y$ (iff $y\in x⋎y$). Then ⪯ is a preorder.

2. 

$x⋏y⪯x$ and $x⋏y⪯y$.

3. 

$x⪯x⋎y$ and $y⪯x⋎y$.

4. 

$x⋏y⪯x⋎y$.

Proof. 

• Since L is a hyperalgebra, there exists some $z\in x⋎x$ which implies (by R3) that $x\in x⋏z$. By R6 we get $x\in x⋎x$. Similarly we conclude $x\in x⋏x$. In particular, $x⪯x$.
  Now, let $x⪯y$ and $y⪯z$. Then $x\in x⋎y$ and $y\in y⋎z$ which implies (by R2) that $x⋎z=x⋎y$. In particular, $x\in x⋎z$ and $x⪯z$.
• Let $z\in x⋏y$. Using R1 and R6 we get $x\in x⋎z=z⋎x$. By R3 we get $z\in z⋏x$, which means that $z⪯x$. Similarly we get $z⪯y$.
• Let $z\in x⋎y$. Using R1 and R3 we get $x\in x⋏z=z⋏x$. By R6 we get $z\in z⋎x$, which means that $x⪯z$. Similarly we get $y⪯z$.
• Just combine items (1)-(3).

□

Example 1.

1. 

Every lattice is a m-hyperlattice.

2. 

Let L be the set of formulas of propositional classical logic over signature $\{\wedge ,\vee ,\to ,\neg \}$. It is well-known (from the fact that propositional classical logic is a Tarskian logic) that the relation in L

$$\alpha ⪯\beta iff\alpha ⊢\beta$$

defines a preorder and so $\langle P,⪯\rangle$ is a proset. Moreover, the sets $\alpha ⋏\beta$ and $\alpha ⋎\beta$ are nonempty for every $\alpha ,\beta \in L$: indeed, $\alpha \wedge \beta \in \alpha ⋏\beta$ and $\alpha \vee \beta \in \alpha ⋎\beta$. This is a consequence of the following facts:

(i) $\alpha \wedge \beta ⊢\alpha$ and $\alpha \wedge \beta ⊢\beta$; and: $\gamma ⊢\alpha$ and $\gamma ⊢\beta$ implies that $\gamma ⊢\alpha \wedge \beta$.

(ii) $\alpha ⊢\alpha \vee \beta$ and $\beta ⊢\alpha \vee \beta$; and: $\alpha ⊢\gamma$ and $\beta ⊢\gamma$ implies that $\alpha \vee \beta ⊢\gamma$.

Then, L is a m-hyperlattice.

3. 

The previous example can be adapted to any propositional Tarskian logic containing (standard) conjunction and disjunction as, for instance, (positive) intuitionistic logic.

4. 

([4, Ch. II, §2, p. 124]) Let V be a vector space (over a field F) and for $A,B\subseteq V$ consider

$$\begin{array}{cc}\hfill A⋎B& :=\{E\subseteq V:\langle E\rangle =\langle A\cup B\rangle \}\hfill \\ \hfill A⋏B& :=\{E\subseteq V:\langle E\rangle =\langle A\rangle \cap \langle B\rangle \},\hfill \end{array}$$

where $\langle E\rangle$ denotes the vector subspace of V generated by $E\subseteq V$. Then $\langle \mathcal{P}\left(V\right),⋎,⋏\rangle$ is a m-hyperlattice.

5. 

The previous example works if we change vector spaces by other algebraic structures: for instance, algebras, modules, rings, abelian groups etc.

6. 

Let $\mathcal{C}$ be a small category with binary products ∏ and coproducts ∐ (see, for instance, [11]) and for $a,b\in Obj\left(\mathcal{C}\right)$ (the set of objects of $\mathcal{C}$), define

$$a⪯biffthereexistamorphismf:a\to bin\mathcal{C}.$$

It is straightforward to check that $\langle Obj\left(\mathcal{C}\right),⪯\rangle$ is a m-hyperlattice. In this case, $a\prod b\in a⋏b$ and $a\coprod b\in a⋎b$.

Definition 7.

The category MHL of hyperlattices is the one where the objects are hyperlattices and the morphism are just the usual morphisms of hyperalgebras. In other words, given $L_1,L_2\in MHL$, a function $f:L_1\to L_2$ is a morphism if for all $x,y,z\in L_1$ we have the following:

1. 

$z\in x⋏y$ implies $f\left(z\right)\in f\left(x\right)⋏f\left(y\right)$;

2. 

$z\in x⋎y$ implies $f\left(z\right)\in f\left(x\right)⋎f\left(y\right)$.

Proposition 1.

Let $L_1,L_2$ be MHLs and $f:L_1\to L_2$ be a function. Then f is a MHL-morphism iff for all $x,y\in L_1$, $x⪯y$ implies $f\left(x\right)⪯f\left(y\right)$.

Proof. 

This is an immediate consequence of the characterization of ⪯ in terms of ⋏ (or ⋎): $x⪯y$ iff $x\in x⋏y$ (iff $y\in x⋎y$). □

Remark 3.

It is worth noting that a version of the celebrated Principle of Duality for ordered sets (see, for instance, [12]) can be easily obtained for prosets, and so for m-hyperlattices. Indeed, if $\mathsf{P}=\langle P,⪯\rangle$ is a proset, so is its dual is ${\mathsf{P}}^{*}=\langle P,⪰\rangle$, where $a⪰b$ iff $b⪯a$, for every $a,b\in P$. Clearly, any statement Φ (just concerning ⪯) has its dual statement${\Phi }^{*}$ (obtained from Φ by replacing ⪯ by ⪰). From this, $\mathsf{P}$ satisfies ${\Phi }^{*}$ iff ${\mathsf{P}}^{*}$ satisfies ${\Phi }^{*}$. Since ${\Phi }^{**}=\Phi$, it follows that Φ holds in any $\mathsf{P}$ iff ${\Phi }^{*}$ holds in any $\mathsf{P}$ (Principle of Duality for prosets). This can be extended to m-hyperlattices, and so if $\mathsf{L}=\langle L,⋏,⋎\rangle$ is a hyperlattice (with underlying preorder ⪯), its dual ${\mathsf{L}}^{*}=\langle L,{⋏}^{*},{⋎}^{*}\rangle$ (with underlying preorder ⪰) is also a m-hyperlattice, where the hyperoperations are defined by $a{⋏}^{*}b:=a⋎b$ and $a{⋎}^{*}b:=a⋏b$, for every $a,b\in L$. By the principle of duality for prosets, and by definition of m-hyperlattices, if a statement Φ (containing ⪯, ⋏, and ⋎) holds in any m-hyperlattice then its dual statement ${\Phi }^{*}$ (obtained from Φ by replacing ⪯ by ⪰, ⋏ by ⋎, and ⋎ by ⋏, respectively) also holds in any m-hyperlattice.

Proposition 2.

Let $\mathsf{P}=\langle P,⪯,⋏,⋎\rangle$ be a m-hyperlattice. If $x,y\in P$ such that $x\equiv y$, then for every $z\in P$ and $B\subseteq P$

1. 

$x⪯z$ iff $y⪯z$.

2. 

$x⪯B$ iff $y⪯B$.

3. 

$x⋎z=y⋎z$.

4. 

$z⪯x$ iff $z⪯y$.

5. 

$B⪯x$ iff $B⪯y$.

6. 

$x⋏z=y⋏z$.

Proof. 

Items (1)-(2) and (4)-(5) follow from the transitivity of ⪯. For (3), observe that $Ub(x,z)=Ub(y,z)$ for every $z\in P$, by (1). Hence

$$x⋎z=\mathsf{Min}\left(Ub\right(x,z\left)\right)=\mathsf{Min}\left(Ub\right(y,z\left)\right)=y⋎z.$$

Item (6) is proved analogously. □

Remark 4.

Given non-empty sets $A,B\subseteq L$, the notation $A⪯B$ established in Definition 4 can be extended to the hyperoperations ⋏ and ⋎ as follows: $A⋏B:=\bigcup \{a⋏b:a\in Aandb\in B\}$, and $A⋎B:=\bigcup \{a⋎b:a\in Aandb\in B\}$.

Proposition 3.

Let $\mathsf{L}$ be an m-hyperlattice, and $x,y,z\in L$. Then:

1. 

If $\varnothing \ne A\subseteq B\subseteq L$ then $A\#B=B\#A$ for $\#\in \{⋏,⋎\}$.

2. 

If $\varnothing \ne A\subseteq B\subseteq L$ then $\mathsf{Max}\left(A\right)⪯\mathsf{Max}\left(B\right)$.

3. 

If $\varnothing \ne A\subseteq B\subseteq L$ then $\mathsf{Min}\left(B\right)⪯\mathsf{Min}\left(A\right)$.

4. 

If $x⪯y$ then $x⋏z⪯y⋏z$.

5. 

If $x⪯y$ then $x⋎z⪯y⋎z$.

6. 

If $\varnothing \ne A,B\subseteq L$ and $A⪯B$ then $A\#c⪯B\#c$ for $\#\in \{⋏,⋎\}$ and any $c\in L$.

7. 

Let $\varnothing \ne B\subseteq L$. If $a⪯x$ and $a⪯B$ then $a⪯x⋏B$.

8. 

Let $\varnothing \ne B\subseteq L$. If $x⪯a$ and $B⪯a$ then $x⋎B⪯a$.

9. 

Let $\varnothing \ne A,B\subseteq L$. If $A⪯z$ and $B⪯z$ then $A⋎B⪯z$.

10. 

Let $\varnothing \ne A,B\subseteq L$. If $x⋏A⪯z$ then $x⋏(A⋏B)⪯z$.

11. 

Let $\varnothing \ne A,B\subseteq L$. If $B⋏y⪯z$ then $(A⋏B)⋏y⪯z$.

12. 

$(x⋏y)⋏z\equiv x⋏(y⋏z)$.

13. 

$(x⋎y)⋎z\equiv x⋎(y⋎z)$.

Proof. 

• It follows from the fact that $A\#B=\bigcup \{a\#b:a\in A,b\in B\}$ and by (R1) and (R4) from Remark 2.
• Assume that $\varnothing \ne A\subseteq B\subseteq L$, and let $x\in \mathsf{Max}\left(A\right)$ and $y\in \mathsf{Max}\left(B\right)$. Since $x\in A$ then $x\in B$, hence $x⪯y$. This shows that $\mathsf{Max}\left(A\right)⪯\mathsf{Max}\left(B\right)$.
• Assume that $\varnothing \ne A\subseteq B\subseteq L$, and let $x\in \mathsf{Max}\left(A\right)$ and $y\in \mathsf{Max}\left(B\right)$. Since $x\in A$ then $x\in B$, hence $x⪯y$. This shows that $\mathsf{Max}\left(A\right)⪯\mathsf{Max}\left(B\right)$.
• Assume that $x⪯y$. By definition, $x⋏z=\mathsf{Max}\left(A\right)$ and $y⋏z=\mathsf{Max}\left(B\right)$, where $A=\{w\in L:w⪯x\mathrm{and}w⪯z\}$ and $B=\{w\in L:w⪯y\mathrm{and}w⪯z\}$. By definition of m-hyperlattices, $\varnothing \ne A\subseteq B\subseteq L$. By item (2), $x⋏z⪯y⋏z$.
• Assume that $x⪯y$. By definition, $x⋎z=\mathsf{Min}\left(A\right)$ and $y⋎z=\mathsf{Min}\left(B\right)$, where $A=\{w\in L:x⪯w\mathrm{and}z⪯w\}$ and $B=\{w\in L:y⪯w\mathrm{and}z⪯w\}$. By definition of m-hyperlattices, $\varnothing \ne B\subseteq A\subseteq L$. By item (3), $x⋎z⪯y⋎z$.
• Fix $\#\in \{⋏,⋎\}$ and $c\in L$, and let $x\in A\#c$, $y\in B\#c$. Then $x\in a\#c$ and $y\in b\#c$ for some $a\in A$ and $b\in B$. By hypothesis, $a⪯b$ and so $a\#c⪯b\#c$, by items (4) and (5). From this, $x⪯y$ and so $A\#c⪯B\#c$.
• Assume that $a⪯x$ and $a⪯B$. If $b\in B$ then $a⪯b$, hence $a⪯x⋏b$. Let $c\in x⋏B$. Hence, $c\in x⋏b$ for some $b\in B$. Since $a⪯x⋏b$, it follows that $a⪯c$. That is, $a⪯x⋏B$.
• A proof can be obtained by dualizing the one given for item (7), i.e.: by changing $t⪯t^{\prime }$ by $t^{\prime }⪯t$, and ⋏ by ⋎.
• Suppose that $A⪯z$ and $B⪯z$, and let $a\in A$. Then, $a⪯z$ and so, by (8), $a⋎B⪯z$, for every $a\in A$. Then $A⋎B=\bigcup \{a⋎b:a\in A,b\in B\}=\bigcup \{a⋎B:a\in A\}⪯z$.
• Assume that $x⋏A⪯z$. Let $w\in x⋏(A⋏B)$. Then $w\in x⋏u$ such that $u\in A⋏B$. Hence, there exists $a\in A$ and $b\in B$ such that $u\in a⋏b$. Since $a⋏b⪯a$ it follows that $u⪯a$. By (4), $x⋏u⪯x⋏a$, and then $w⪯x⋏a$ (given that $w\in x⋏u$). Since $x⋏a\subseteq x⋏A⪯z$ it follows that $x⋏a⪯z$ and so $w⪯z$. That is, $x⋏(A⋏B)⪯z$.
• It follows from (10) and by commutativity of ⋏, namely: $A⋏B=B⋏A$ and $A⋏x=x⋏A$ for every A and x.
• Let $a\in (x⋏y)⋏z$. Then, $a\in b⋏z$ for some $b\in x⋏y$. From $a\in b⋏z⪯b$ we infer that $a⪯b$. From $b\in x⋏y⪯x$ it follows that $b⪯x$. By transitivity, $a⪯x$ (*). Now, $b\in x⋏y⪯y$ and so $b⪯y$. By (4), $b⋏z⪯y⋏z$. But $a\in b⋏z$, hence $a⪯y⋏z$ (**). From (*) and (**) we prove that $a⪯x⋏(y⋏z)$, by (7). This shows that $(x⋏y)⋏z⪯x⋏(y⋏z)$.
  Conversely, let $a\in x⋏(y⋏z)$. Then, $a\in x⋏b$, for some $b\in y⋏z$. Since $b\in y⋏z⪯y$ it follows that $b⪯y$ and so $x⋏b⪯x⋏y$, by (4). But then $a\in x⋏b⪯x⋏y$, which implies that $a⪯x⋏y$ (@). In turn, $a\in x⋏b⪯b$, hence $a⪯b$. Since $b\in y⋏z⪯z$ it follows that $b⪯z$, and then $a⪯z$ (@@). By (@), (@@) and (7) (and by the fact that $B⋏w=w⋏B$) it follows that $a⪯(x⋏y)⋏z$. That is, $x⋏(y⋏z)⪯(x⋏y)⋏z$.
• A proof can be obtained by dualizing the one given for item (12), i.e.: by changing $t⪯t^{\prime }$ by $t^{\prime }⪯t$, and ⋏ by ⋎, and now by using (5) and (8) instead of (4) and (7), respectively.

□

Remark 5.

1. 

Let $\varnothing \ne A,B\subseteq L$. It is worth noting that $A⪯B$ does not imply, in general, that $A\#C⪯B\#C$ for $\#\in \{⋏,⋎\}$ and $\varnothing \ne C\subseteq L$. Moreover: $a⪯b$ does not imply (in general) that $a\#C⪯b\#C$ for $\#\in \{⋏,⋎\}$ and $\varnothing \ne C\subseteq L$. This shows that item (6) of Proposition 3 cannot be generalized to arbitrary non-empty sets C. To see an example, recall the m-hyperlattice L considered in Example 1 (2). Let $p,q,r,s$ be 4 different propositional variables, and let $a=p\wedge q$, $b=p$, and $C=\{r,s\}$. Clearly, $a⪯b$. However, $a⋏C⪱b⋏C$. Indeed, $(p\wedge q)\wedge r\in a⋏C$ and $p\wedge s\in b⋏C$ but $(p\wedge q)\wedge r⪱p\wedge s$. Analogously, $a⪯b$ but $a⋎C⪱b⋎C$: indeed, $(p\wedge q)\vee r\in a⋎C$ and $p\vee s\in b⋎C$ but $(p\wedge q)\vee r⪱p\vee s$. In order to guarantee monotonicity for ⋏ and ⋎ w.r.t. sets, stability (see Definition 8) is required, as we shall see below in Proposition 5.

2. 

It can be proven that, in general, $A⋏B⪱A$, for $\varnothing \ne A,B\subseteq L$. Moreover, $A⋏x⪱A$ in general, for $A\ne \varnothing$. Analogously, $A⪱A⋎x$ and so $A⪱A⋎B$ in general. Examples can be found, once again, in the m-hyperlattice of Example 1 (2). Indeed, take $A=\{p,q\}$ and $x=r$ for $p,q,r$ different propositional variables. Since $p\wedge r⪱q$ then $\{p,q\}⋏r⪱\{p,q\}$. Analogously, since $q⪱p\vee r$ then $\{p,q\}⪱\{p,q\}⋎r$. In order to guarantee the validity of these desirable properties, it is required stability once again, see Proposition 4 below.

Definition 8.

Let L be a m-hyperlattice, and let $\varnothing \ne A,B\subseteq L$. We say that A and B are similar, and write $A\equiv B$, if $a\equiv b$ for every $a\in A$ and $b\in B$. That is: $a⪯b$ and $b⪯a$ for every $a\in A$ and $b\in B$. A non-empty subset $A\subseteq L$ is stable if $A\equiv A$.

Proposition 4.

Let $A,B\subseteq L$ be non-empty stable subsets. Then $A⋏B$ and $A⋎B$ are stable subsets. In particular, for all $a\in A$ and $b\in B$ we have

$$A⋏B=a⋏b \mathrm{and} A⋎B=a⋎b.$$

Proof. 

Let $x,y\in A⋏B$. Then $x\in a⋏b$ and $y\in a^{\prime }⋏b^{\prime }$ for some $a,a^{\prime }\in A$ and $b,b^{\prime }\in B$. Since $a\equiv a^{\prime }$ and $b\equiv b^{\prime }$, (6) of Proposition 2 provides $a⋏b=a^{\prime }⋏b^{\prime }$ and we have

$$x,y\in a⋏b=\mathrm{Max}\left(\mathrm{Lb}\right(a,b\left)\right).$$

Since x is a lower bound of a and b, we have $x⪯a$ and $x⪯b$, providing $x⪯⋏b$, which imply $x⪯y$. Similarly we get $y⪯x$, providing $x\equiv y$. Hence $A⋏B$ is stable. For the final part, we already know that if $a,a^{\prime }\in A$ and $b,b^{\prime }\in B$ then $a⋏a^{\prime }\equiv b⋏b^{\prime }$ (Proposition 2 item (6) again). Then for all $x\in A$ and $y\in B$ we get

$$A⋏B=\bigcup _{a\in A,b\in B}a⋏b=x⋏y.$$

The proof for $A⋎B$ follows by duality. □

Proposition 5.

Let $A,B,C,D\subseteq L$ be stable sets such that $A⪯B$ and $C⪯D$. Then $A⋏C⪯B⋏D$ and $A⋎C⪯B⋎D$.

Proof. 

Since $A,B,C,D$ are stable, we can suppose by Proposition 4 that $A⋏C=a⋏c$ and $B⋏D=b⋏d$ for some $a\in A$, $b\in B$, $c\in C$ and $d\in D$ such that, by hypothesis, $a⪯b$ and $c⪯d$. Since $a⋏c⪯a⪯b$ and $a⋏c⪯c⪯d$, we get $a⋏c⪯a⪯b⋏d$, providing that $A⋏C=a⋏c⪯b⋏d=B⋏D$. The proof for $A⋎B$ is analogous. □

Supremoinds and infimoids between stable sets can be characterized in a natural way:

Proposition 6.

Let $A,B\subseteq L$ be stable sets. Then $A⋏B=\mathsf{Max}\left(Lb\right(A\cup B\left)\right)$ and $A⋎B=\mathsf{Min}\left(Ub\right(A\cup B\left)\right)$.

Proof. 

Let $a\in A$ and $b\in B$. By stability of both A and B, it is immediate to see that $Lb(A\cup B)=Lb\left(\right\{a,b\left\}\right)$. From this and by Proposition 4, $A⋏B=a⋏b=\mathsf{Max}\left(Lb\right(\{a,b\}\left)\right)=\mathsf{Max}\left(Lb\right(A\cup B\left)\right)$. The proof for ⋎ follows by duality. □

As it would be expected, the absorption laws of lattices hold for m-hyperlattices in a suitable weaker form:

Proposition 7 (Absorption laws for m-hyperlattices).

Let L be a m-hyperlattice, and let $x,y\in L$. Then:

$$x⋏(x⋎y)\equiv x\equiv x⋎(x⋏y).$$

Proof. 

Let $A=x⋏(x⋎y)$. Since $\left\{x\right\}$ and $x⋎y$ are stable, $A=\mathsf{Max}\left(Lb\right(\left\{x\right\}\cup (x⋎y)\left)\right))$, by Proposition 6. Since $x⪯x$ and $x⪯x⋎y$, it follows that $x⪯A$. Conversely, let $a\in A$. Then $a⪯x$, since $a\in Lb\left(\right\{x\}\cup (x⋎y\left)\right)$. This shows that $A⪯x$ and so $A\equiv x$. The rest of the proof follows by duality. □

Proposition 8.

Let $A,B,C\subseteq L$ be stable subsets. Then, for all $a\in A$, $b\in B$ and $c\in C$ it holds:

$$\begin{array}{cc}\hfill A⋏(B⋏C)& =a⋏(b⋏c),\hfill \\ \hfill (A⋏B)⋏C& =(a⋏b)⋏c,\hfill \\ \hfill A⋎(B⋎C)& =a⋎(b⋎c),\hfill \\ \hfill (A⋎B)⋎C& =(a⋎b)⋎c,\hfill \\ \hfill (A⋏B)⋎C& =(a⋏b)⋎c,\hfill \\ \hfill (A⋎B)⋏C& =(a⋎b)⋏c.\hfill \end{array}$$

In particular,

$$A⋏(B⋏C)\equiv (A⋏B)⋏C and A⋎(B⋎C)\equiv (A⋎B)⋎C.$$

Proof. 

Given $d\in b⋏c$, by Proposition 4 we conclude that

$$A⋏(B⋏C)=A⋏(b⋏c)=a⋏d.$$

Since $b⋏c$ is stable and for $d,d^{\prime }\in b⋏c$ we have $a⋏d=a⋏d^{\prime }$, we get $A⋏(B⋏C)=a⋏(b⋏c)$. The proof for the other cases is similar. □

Definition 9.

Let $\mathsf{P}=\langle P,⪯,⋏,⋎\rangle$ be a m-hyperlattice. The sets $\mathsf{Min}\left(P\right)$ and $\mathsf{Max}\left(P\right)$ of minima and maxima elements of P will be denoted by ⊥ (or 0) and ⊤ (or 1), respectively. We say that a m-hyperlattice P is bounded if $\top \ne \varnothing \ne \perp$.

Remark 6.

Let $\varnothing \ne A\subseteq P$. If $\top \ne \varnothing$, then $A\equiv \top$ iff $A\subseteq \top$. Analogously, if $\perp \ne \varnothing$, then $A\equiv \perp$ iff $A\subseteq \perp$.

To end this section, it will be shown that the denotation, in any concrete hyperlattice L, of any term in the signature of m-hyperlattices, produces a stable subset of L.

Definition 10 (M-term).

Let L be a m-hyperlattice. We define M-terms recursively as follows:

1. 

Every non-empty stable subset $A\subseteq L$ is a M-term.

2. 

If A and B are M-terms, then $A⋏B$ and $A⋎B$ are M-terms.

Proposition 9.

Let $A\subseteq L$ be a M-term. Then A is a stable subset.

Proof. 

Just use induction and Propositions 4 and 8. □

3. Sette Implicative Hyperlattices

In this section we recall the class of hyperalgebras introduced by Sette in [8] (that we called Sette implicative hyperlattices) as a basis of the hyperalgebraic semantics for da Costa logic $C_{\omega }$ he also proposed. The intuition behind this is to generalize the notion of implicative lattices to the context of hyperalgebras. As we shall see in Proposition 10, Sette’s intuitions were right.

Definition 11 (Sette implicative hyperlattices, [8, Definition 2.3]).

A Sette implicative hyperlattice (or a SIHL)) is a hyperalgebra $\mathsf{L}=\langle L,⋏,⋎,⊸\rangle$ such that the reduct $\langle L,⋏,⋎\rangle$ is a m-hyperlattice and the hyperoperator ⊸ satisfies the following properties, for every $x,y,z,z^{\prime }\in L$:

(I1) 

$z\in x⊸y$ implies that $x⋏z⪯y$;

(I2) 

$x⋏z⪯y$ implies that $z⪯x⊸y$;

(I3) 

$z\equiv z^{\prime }$ and $z\in x⊸y$ implies that $z^{\prime }\in x⊸y$.

It is possible to give a more direct characterization of SIHLs.

Definition 12.

Let $\mathsf{P}$ be a m-hyperlattice, and let $x,y\in P$. The set $\mathsf{R}(x,y)$ is given by $\mathsf{R}(x,y)=\{z\in P:x⋏z⪯y\}$.

Proposition 10.

Let $\mathsf{L}=\langle L,⋏,⋎,⊸\rangle$ be a hyperalgebra such that $\langle L,⋏,⋎\rangle$ is a m-hyperlattice. Then, $\mathsf{L}$ is an SIHL iff $x⊸y=\mathsf{Max}\left(\mathsf{R}\right(x,y\left)\right)$, for every $x,y\in L$.

Proof. 

‘Only if’ part: Suppose that $\mathsf{L}$ is an SIHL, and let $z\in x⊸y$. By (I1), $x⋏z⪯y$ and so $z\in \mathsf{R}(x,y)$. Now, let $z^{\prime }\in \mathsf{R}(x,y)$. Then, $x⋏z^{\prime }⪯y$ which implies that $z^{\prime }⪯x⊸y$, by (I2). Since $z\in x⊸y$, by hypothesis, then $z^{\prime }⪯z$. This proves that $\mathsf{R}(x,y)⪯z$ and so $z\in \mathsf{Max}\left(\mathsf{R}\right(x,y\left)\right)$. That is, $x⊸y\subseteq \mathsf{Max}\left(\mathsf{R}\right(x,y\left)\right)$. In order to prove the converse inclusion, let $z\in \mathsf{Max}\left(\mathsf{R}\right(x,y\left)\right)$. Then, $z\in \mathsf{R}(x,y)$ such that $\mathsf{R}(x,y)⪯z$. It holds that $x⊸y\subseteq \mathsf{R}(x,y)$, by (I1), then

$(\ast )x⊸y⪯z$.

In turn, from $z\in \mathsf{R}(x,y)$ it follows that $x⋏z⪯y$ and so

$(\ast \ast )z⪯x⊸y$,

by (I2). Now, let $z^{\prime }\in x⊸y$ (observe that $x⊸y\ne \varnothing$, given that $\mathsf{L}$ is a hyperalgebra). By $(\ast )$ and $(\ast \ast )$, $z^{\prime }⪯z$ and $z⪯z^{\prime }$, that is, $z\equiv z^{\prime }$. By (I3), $z\in x⊸y$. That is, $\mathsf{Max}\left(\mathsf{R}\right(x,y\left)\right)\subseteq x⊸y$.

‘If’ part: Let $x⊸y:=\mathsf{Max}\left(\mathsf{R}\right(x,y\left)\right)$, for every $x,y\in L$. Since $\mathsf{L}$ is an hyperalgebra, by hypothesis, then $x⊸y\ne \varnothing$, for every $x,y\in L$. If $z\in x⊸y$ then, by definition of ⊸, $z\in \mathsf{R}(x,y)$, which means that $x⋏z⪯y$. Hence, ⊸ satisfies (I1). Suppose now that $x⋏z⪯y$. Then $z\in \mathsf{R}(x,y)$, therefore $z⪯\mathsf{Max}\left(\mathsf{R}\right(x,y\left)\right)=x⊸y$. This shows that ⊸ satisfies (I2). Finally, suppose that $z\equiv z^{\prime }$ and $z\in x⊸y$. By Proposition 2 (6), and given that $z\in \mathsf{R}(x,y)$, it follows that $x⋏z^{\prime }=x⋏z⪯y$. That is, $z^{\prime }\in \mathsf{R}(x,y)$. Since $\mathsf{R}(x,y)⪯z$, it follows from Proposition 2 (5) that $\mathsf{R}(x,y)⪯z^{\prime }$. This implies that $z^{\prime }\in \mathsf{Max}\left(\mathsf{R}(x,y)\right)=x⊸y$. That is, ⊸ also satisfies (I3). □

Remark 7.

The latter proposition shows that Sette’s intuitions when defining implicative hyperlattices based on Morgado hyperlattices were right: indeed, $x⊸y=\mathsf{Max}\left(\mathsf{R}\right(x,y\left)\right)$ seems to be the more natural generalization of the notion of implicative lattices to the realm of Morgado hyperlattices.

Corollary 1.

Let $\mathsf{L}=\langle L,⋏,⋎\rangle$ be a m-hyperlattice. If $\mathsf{Max}\left(\mathsf{R}\right(x,y\left)\right)\ne \varnothing$ for every $x,y\in L$ then $\mathsf{L}$ is a SIHL such that $x⊸y=\mathsf{Max}\left(\mathsf{R}\right(x,y\left)\right)$, for every $x,y\in L$.

Lemma 2.

Let $\mathsf{L}$ be a m-hyperlattice. Then:

1. 

$y\in \mathsf{R}(x,y)$.

2. 

For all $x,y\in L$ there exist $z\in x⊸y$ such that $y⪯z$.

3. 

If $y\equiv y^{\prime }$ then $R(x,y)=R(x,y^{\prime })$. In particular, $x⊸y=x⊸y^{\prime }$.

4. 

If $x\equiv x^{\prime }$ then $R(x,y)=R(x^{\prime },y)$. In particular, $x⊸y=x^{\prime }⊸y$.

5. 

If $x\equiv x^{\prime }$ and $y\equiv y^{\prime }$ then $R(x,y)=R(x^{\prime },y^{\prime })$. In particular, $x⊸y=x^{\prime }⊸y^{\prime }$.

6. 

If $z,z^{\prime }\in R(x,y)$ then $z⋏z^{\prime }\subseteq R(x,y)$.

7. 

$R(x,y)$ is stable.

8. 

If $z\in R(x,y)$ then $R(x,z)\subseteq R(x,y)$. In particular, $x⊸y\subseteq x⊸z$.

9. 

$x⊸y$ is stable.

10. 

For all $x,y,z\in L$, $z⪯x⊸y$ iff $z⋏x⪯y$.

Proof. 

• It follows from $x⋏y⪯y$.
• It follows from $y\in R(x,y)$ and $x⊸y=Max\left(R\right(x,y\left)\right)$.
• If $z\in R(x,y)$, then $x⋏z⪯y⪯y^{\prime }$, which means $z\in R(x,y^{\prime })$. Similarly $z\in R(x,y)$.
• Proposition 4 provides that $x⋏z\equiv x^{\prime }⋏z$ for all $z\in L$. If $z\in R(x,y)$, then $x⋏z⪯y$ and since $x\equiv x^{\prime }$, we get $x^{\prime }⋏z⪯y$, providing $z\in R(x^{\prime },y)$. Similarly $z\in R(x^{\prime },y)$ imply $z\in R(x,y)$. Therefore $R(x,y)=R(x^{\prime },y)$ and in particular, $x⊸y=x^{\prime }⊸y$.
• Just note that combining the previous items we get

  $$R(x,y)=R(x^{\prime },y)=R(x^{\prime },y^{\prime }).$$
• It follows from $x⋏(z⋏z^{\prime })⪯x⋏z$ and $x⋏(z⋏z^{\prime })⪯x⋏z^{\prime }$.
• It follows from the fact that $z⪯z^{\prime }$ implies $x⋏z⪯x⋏z^{\prime }$.
• Let $z\in R(x,y)$ and $z^{\prime }\in R(x,z)$. Then $x⋏z⪯y$ and $x⋏z^{\prime }⪯z$. From $x⋏z^{\prime }⪯z$ and $x⋏z^{\prime }⪯x$ we get $x⋏z^{\prime }⪯x⋏z⪯y$, providing $z^{\prime }\in R(x,y)$.
• Let $z,z^{\prime }\in x⊸y$. From $z\in \mathrm{Ub}\left(R\right(x,y\left)\right)$ and $z^{\prime }\in x⊸y=\mathrm{Max}\left(\mathrm{Ub}\left(R(x,y)\right)\right)$ we get $z^{\prime }⪯z$. Similarly $z⪯z^{\prime }$.
• Let $z⪯x⊸y$. Then $z⪯z^{\prime }$ for all $z^{\prime }\in x⊸y$. By item (4) of Proposition 3 we get $x⋏z⪯x⋏z^{\prime }$ and by (I1), $x⋏z^{\prime }⪯y$ establishing $x⋏z⪯y$. The converse is an immediate consequence of (I2).

□

Proposition 11.

Let $A,B,C,D\subseteq L$ be stable subsets. Then

$$A⋏(B⋎C)\equiv (A⋏B)⋎(A⋏C) and A⋎(B⋏C)\equiv (A⋎B)⋏(A⋎C).$$

Proof. 

By an argument similar to the used in Proposition 8 we get

$$\begin{array}{cc}\hfill A⋏(B⋎C)& =a⋏(b⋎c)\hfill \\ \hfill (A⋏B)⋎(A⋏C)& =(a⋏b)⋎(a⋏c)\hfill \end{array}$$

for all $a\in A$, $b\in B$ and $c\in C$. Since by 2 we have $a⋏(b⋎c)\equiv (a⋏b)⋎(a⋏c)$, we get $A⋏(B⋎C)\equiv (A⋏B)⋎(A⋏C)$. The proof for the other equation is similar. □

We finish this Section by proving some basic but useful properties of SIHLs. They will be used in Section 6.

Proposition 12.

Let $\mathsf{L}$ be a SIHL, and let $x,y,z\in L$. Then:

1. 

$x⋏(x⊸y)⪯y$.

2. 

$x⪯y$ iff $x⊸y=\top$.

3. 

$x⊸x=\top$.

4. 

$x⊸(y⊸x)=\top$.

5. 

$x⊸(y⊸(x⋏y\left)\right)=\top$.

6. 

$(x⋏y)⊸x=\top$ and $(x⋏y)⊸y=\top$.

7. 

$x⊸(x⋎y)=\top$ and $y⊸(x⋎y)=\top$.

Proof. 

• Let $w\in x⋏(x⊸y)=\bigcup \{x⋏z:z\in (x⊸y\left)\right\}$. Then, $w\in x⋏z$ for some $z\in (x⊸y)$. But $z\in \mathsf{R}(x,y)$ and so $x⋏z⪯y$. From this, $w⪯y$.
• Suppose that $x⪯y$, and let $z\in L$. Given that $x⋏z⪯x$ then $x⋏z⪯y$. That is, $\mathsf{R}(x,y)=L$. From this, $x⊸y=\mathsf{Max}\left(\mathsf{R}\right(x,y\left)\right)=\mathsf{Max}\left(L\right)=\top$. Conversely, suppose that $x⊸y=\mathsf{Max}\left(\mathsf{R}\right(x,y\left)\right)=\top$, and let $z\in x⊸y$. Then, $x⋏z⪯y$. But $x⪯z$ (since $z\in \top$), then $x\in x⋏z$ (recall Remark 2). This implies that $x⪯y$.
• It follows from (2) and the fact that $x⪯x$.
• There exist $z\in y⊸x$ such that $x⪯z$. Using (2) we get

  $$\top =x⊸z\subseteq x⊸(y⊸x).$$
• It follows from (2) and the fact that $x,y\in R(y,x⋏y)$ (so there exist $z\in y⊸(x⋏y)$ with $x⪯z$ and $y⪯z$).
• It follows from (2) and the fact that $x,y\in R(x,x⋏y)$ (and also $x,y\in R(y,x⋏y)$).
• It follows from (2) and the fact that $x,y\in R(x,x⋎y)$ (and also $x,y\in R(y,x⋎y)$).

□

Proposition 13.

Let $A,B\subseteq L$ be stable non-empty subsets. Then for all $a\in A$ and all $b\in B$,

$$A⊸B=a⊸b.$$

In particular, $A⊸B$ is stable.

Proof. 

Follows from Lemma 2 items (5) and (9). □

Proposition 14.

Let $\mathsf{L}$ be a SIHL. Let $A,B,C\subseteq L$ such that $A\ne \varnothing$, $B\ne \varnothing$ and $C\ne \varnothing$. Let $x,y,z\in L$. Then:

1. 

$A⪯B$ iff $a⊸b=\top$ for every $a\in A$ and $b\in B$, iff $A⊸B=\top$. In particular, $A⪯B$ iff $a⊸b=\top$ for some $a\in A$ and some $b\in B$.

2. 

$x⪯y⊸z$ iff $x⋏y⪯z$.

3. 

$A⪯B⊸C$ iff $a⪯b⊸c$ for every $a\in A$, $b\in B$ and $c\in C$ iff $a⋏b⪯c$ for every $a\in A$, $b\in B$ and $c\in C$ iff $A⋏B⪯C$.

4. 

$A⋏(A⊸B)⪯B$.

5. 

$x⋏(x⊸y)⪯y$.

Proof. 

• Assume that $A⪯B$, and let $a\in A$ and $b\in B$. Since $a⪯b$, by hypothesis, we infer that $a⊸b=\top$, by Proposition 12 (2). Now, suppose that $a⊸b=\top$ for every $a\in A$ and $b\in B$. Then, $A⊸B=\bigcup \{x⊸y:x\in A,y\in B\}=\top$. Finally, suppose that $A⊸B=\top$, and let $a\in A$ and $b\in B$. Then $a⊸b\subseteq \top$. Let $z\in a⊸b$. Then, $a⋏z⪯b$, and $z\in \top$, hence $a⪯z$. From this, $a\in a⋏z$ (by Remark 2) and so $a⪯b$. This shows that $A⪯B$.
• Suppose that $x⪯y⊸z$, and let $w\in y⊸z=\mathsf{Max}\left(\mathsf{R}\right(y,z\left)\right)$. By hypothesis, $x⪯w$ and so, by Proposition 3 (4), $x⋏y⪯w⋏y⪯z$ (since $w\in \mathsf{R}(y,z)$). The converse follows from (I2) of definition of SIHLs.
• Assume that $A⪯B⊸C$ and let $a\in A$, $b\in B$ and $c\in C$. Since $b⊸c\subseteq B⊸C$ then $a⪯b⊸c$ and so, by (2), $a⋏b⪯c$, for every $a,b,c$, Hence $A⋏B⪯c$ for every $c\in C$ and so $A⋏B⪯C$. Conversely, assume that $A⋏B⪯C$ and let $a\in A$, $b\in B$ and $c\in C$. Then $a⋏b⪯c$ (since $a⋏b\subseteq A⋏B$) and so $a⪯b⊸c$, by (2), for every $a,b,c$. Hence $a⪯B⊸C$ for every $a\in A$ and so $A⪯B⊸C$.
• Clearly $A⊸B⪯A⊸B$. By (3), $(A⊸B)⋏A⪯B$, and so $A⋏(A⊸B)⪯B$, by Proposition 3 (1).
• It follows from (4) by taking $A=\left\{x\right\}$ and $B=\left\{y\right\}$. A direct proof is as follows: let $w\in x⋏(x⊸y)=\bigcup \{x⋏z:z\in (x⊸y\left)\right\}$. Then, $w\in x⋏z$ for some $z\in (x⊸y)$. But $z\in \mathsf{R}(x,y)$ and so $x⋏z⪯y$. From this, $w⪯y$.

□

It is well-known that every implicative lattice is distributive; in particular, any Heyting algebra (which is nothing else than an implicative lattice with a bottom element) is distributive. By considering a suitable notion of distributive m-hyperlattices, it will be proven that the same results hold for implicative hyperlattices (see Corollary 2 below).

Definition 13.

A m-hyperlattice L is said to be distributive if, for all $x,y,z\in L$,

$$x⋏(y⋎z)\equiv (x⋏y)⋎(x⋏z)andx⋎(y⋏z)\equiv (x⋎y)⋏(x⋎z).$$

This relatively weak notion of distributive hyperlattices will be enough for proving, in Section 6, the soundness of some important axioms of positive intuitionistic logic w.r.t. implicative hyperlattices (see proof of Theorem 1, $\left(1\right)\Rightarrow \left(2\right)$).

Proposition 15.

In any SIHL the following holds, for any $x,y,z\in L$:

1. 

$x⋏(y⋎z)\equiv (x⋏y)⋎(x⋏z)$.

2. 

$(x⊸(y⊸z\left)\right)⊸\left(\right(x⊸y)⊸(x⊸z\left)\right)=\top$.

3. 

$(x⊸y)⊸\left(\right(y⊸z)⊸(x⋎y)⊸z))=\top$.

4. 

$(x⊸y)⋏(y⊸z)⪯(x⋎y)⊸z)$.

5. 

If A is stable then: $A⋏x⪯z$ and $A⋏y⪯z$ implies that $A⋏(x⋎y)⪯z$.

6. 

$x⋎(y⋏z)\equiv (x⋎y)⋏(x⋎z)$.

Proof. 

• Recall Proposition 6, and let

  $$w\in (x⋏y)⋎(x⋏z)=\mathrm{Min}\left(Ub\right((x⋏y)\cup (x⋏z)\left)\right).$$
  Since $x⋏y⪯x⋏(y⋎z)$ and $x⋏z⪯x⋏(y⋎z)$, we have that

  $$x⋏(y⋎z)\subseteq \mathrm{Ub}\left(\right(x⋏y)\cup (x⋏z\left)\right)).$$
  This implies that $w⪯x⋏(y⋎z)$ and so $(x⋏y)⋎(x⋏z)⪯x⋏(y⋎z)$.
  In order to prove the other inequality, let

  $$w\in (x⋏y)⋎(x⋏z)=\mathrm{Min}\left(Ub\right((x⋏y)\cup (x⋏z)\left)\right).$$
  Then, $x⋏y⪯w$ and $x⋏z⪯w$. From this $y⪯x⊸w$ and $z⪯x⊸w$, by (I2), which implies that $y⋎z⪯x⊸w$. Now, let $u\in y⋎z$. Then $u⪯x⊸w$, which implies by item (10) of Proposition 2 that $x⋏u⪯w$. By Proposition 4, the latter implies that $x⋏(y⋎z)=x⋏u⪯w$. Therefore, $x⋏(y⋎z)⪯(x⋏y)⋎(x⋏z)$.
• By Proposition 14 (3), the following holds:

  $$\begin{array}{cc}\hfill & (x⊸(y⊸z\left)\right)⊸\left(\right(x⊸y)⊸(x⊸z\left)\right)=\top \mathrm{iff}\hfill \\ \hfill & x⊸(y⊸z)⪯(x⊸y)⊸(x⊸z) \mathrm{iff}\hfill \\ \hfill & (x⊸(y⊸z\left)\right)⋏(x⊸y)⪯x⊸z \mathrm{iff}\hfill \\ \hfill & \left[\right(x⊸(y⊸z))⋏(x⊸y\left)\right]⋏x⪯z\hfill \end{array}$$
  Let $A=x⊸(y⊸z)$ and $B=x⊸y$. Observe that A and B are stable, and we have that $B⋏x⪯y$, by Proposition 12 (1), and $y⋏(x⋏A)⪯z$. Indeed, $x⋏A⪯y⊸z$, by Proposition 14 (4), hence $y⋏(x⋏A)⪯z$, by Proposition 14 (3).
  Since $B⋏x⪯y$ and $y⋏(x⋏A)⪯z$, we get (Proposition 5) that $(B⋏x)⋏(x⋏A)⪯z$. Now, note that $x\equiv x⋏x$ and $(B⋏x)⋏(x⋏A)\equiv (A⋏B)⋏x⋏x$, providing that $(A⋏B)⋏x⪯z$.
• By Proposition 14 (3), the following holds:

  $$(x⊸y)⊸\left(\right(y⊸z)⊸(x⋎y)⊸z))=\top \mathrm{iff}A⋏(x⋎y)⪯z,$$

  where $A=(x⊸y)⋏(y⊸z)$. Let $w\in A$. By Proposition 8, $A⋏(x⋎y)=w⋏(x⋎y)$, since A is stable. By Proposition 4, $A⋏x=w⋏x$ and $A⋏y=w⋏y$, hence $(A⋏x)⋎(A⋏y)=(w⋏x)⋎(w⋏y)$. By item (1),

  $$A⋏(x⋎y)=w⋏(x⋎y)\equiv (w⋏x)⋎(w⋏y)=(A⋏x)⋎(A⋏y).$$
  From this, it is required to prove that $(A⋏x)⋎(A⋏y)⪯z$. By Proposition 3 (9), is enough to prove that $A⋏x⪯z$ and $A⋏y⪯z$.
  Since $A⋏y\equiv (x⊸y)⋏[y⋏(y⊸z\left)\right]$ and $y⋏(y⊸z)⪯z$, we get $A⋏y⪯z$. For the other part, note that $A⋏x\equiv [x⋏(x⊸y\left)\right]⋏(y⊸z)$ and then

  $$A⋏x\equiv [x⋏(x⊸y\left)\right]⋏(y⊸z)⪯y⋏(y⊸z)⪯z.$$
• It is an immediate consequence of the previous item (3), and items (1) and (3) of Proposition 14.
• Assume that $A⋏x⪯z$ and $A⋏y⪯z$, and let $w\in A$. Then, $w⋏x⪯z$ and $w⋏y⪯z$, which implies that $w⪯x⊸z$ and $w⪯y⊸z$, by (I2). From this, $w⪯(x⊸z)⋏(y⊸z)$. By item (4) and transitivity, $w⪯(x⋎y)⊸z$. Since A is stable, this shows that $A⪯(x⋎y)⊸z$. By Proposition 14 (3), $A⋏(x⋎y)⪯z$.
• For the proof of $x⋎(y⋏z)\equiv (x⋎y)⋏(x⋎z)$, note that $x⪯x⋎y$ and $y⋏z⪯x⋎z$ provide that $x⋎(y⋏z)⪯(x⋎y)⋏(x⋎z)$. For the other inequality, observe that $x⋏z⪯x⋎(y⋏z)$, and $y⋏z⪯x⋎(y⋏z)$. By item (5) (taking $A=\left\{z\right\}$) it follows that $(x⋎y)⋏z⪯x⋎(y⋏z)$. But clearly $(x⋎y)⋏x⪯x⋎(y⋏z)$, and so $(x⋎y)⋏(x⋎z)⪯x⋎(y⋏z)$, by item (5) once again.

□

Corollary 2.

Every SIHL is a distributive m-hyperlattice.

Proof. 

It follows from items (1) and (6) of Proposition 15. □

Now, Proposition 9 will be extended to SIHLs, showing that the denotation, in any concrete SIHL L, of any term in the signature of SIHLs, produces a stable subset of L.

Definition 14 (S-term).

Let L be a SIHL. We define S-terms recursively as follows:

1. 

Every stable non-empty subset $A\subseteq L$ is an S-term.

2. 

If A and B are S-terms, then $A⋏B$, $A⋎B$ and $A⊸B$ are S-terms.

Proposition 16.

Let $A\subseteq L$ be an S-term. Then A is a stable subset.

Proof. 

Just use induction and Propositions 4, 8 and 13. □

Proposition 17.

Let $A,B,C,D\subseteq L$ be non-empty stable subsets such that $A⪯B$ and $C⪯D$. Then $A⊸C⪯B⊸D$.

Proof. 

We only need to prove that $a⊸c⪯b⊸d$ for some $a\in A$, $b\in B$, $c\in C$ and $d\in D$. Let $z\in R(a,c)$. Then $a⋏z⪯c⪯d$ and $a⋏z⪯a⪯b$, which means $a⋏z⪯b⋏d⪯d$ and $z\in R(b,d)$. Hence, $R(a,c)⪯R(b,d)$ which implies that

$$a⊸c=\mathrm{Max}\left(R\right(a,c\left)\right)⪯\mathrm{Max}\left(R\right(b,d\left)\right)=b⊸d.$$

□

4. The Logic ${\mathbf{C}}_{\mathbf{\omega }}$

Among the most influential contributions of the pioneering work of the Brazilian mathematician Newton da Costa (1929–2024) is the development of the logic $C_{\omega }$, a system that belongs to his well-known hierarchy of paraconsistent logics $C_n$ (for $1\le n\le \omega$) introduced in [13]. $C_{\omega }$ logic, as well as the other systems $C_n$, is defined over the signature ${\Sigma }_{\omega }=\{\wedge ,\vee ,\to ,\neg \}$. The original idea of da Costa was to consider $C_{\omega }$ as the syntactic limit of the hierarchy $C_n$, for $1\le n<\omega$. In fact, the Hilbert calculus for $C_{\omega }$ contains exactly all the axioms that belong to any $C_n$, for $1\le n<\omega$. However, as shown in [14], $C_{\omega }$ it is not the deductive limit of these calculi. Despite this, $C_{\omega }$ has several interesting features: it is based on positive intuitionistic logic instead of positive classical logic, as happens with $C_n$ for $n<\omega$. Indeed, while the latter satisfy Peirce’s law $\left(\right(\alpha \to \beta )\to \alpha )\to \alpha$, the former do not (see [15, Theorem 15]). On the other hand, each $C_n$ (for $n<\omega$) is finitely trivializable, while $C_{\omega }$ is not (see [15, Theorem 8]).

Definition 15 (Hilbert calculus for $C_{\omega }$).

The Hilbert calculus for $C_{\omega }$ over ${\Sigma }_{\omega }$ is defined as follows:

Axiom schemas:

(AX1) 

$\alpha \to (\beta \to \alpha )$

(AX2) 

$(\alpha \to (\beta \to \gamma \left)\right)\to \left(\right(\alpha \to \beta )\to (\alpha \to \gamma \left)\right)$

(AX3) 

$\alpha \to (\beta \to (\alpha \wedge \beta \left)\right)$

(AX4) 

$(\alpha \wedge \beta )\to \alpha$

(AX5) 

$(\alpha \wedge \beta )\to \beta$

(AX6) 

$\alpha \to (\alpha \vee \beta )$

(AX7) 

$\beta \to (\alpha \vee \beta )$

(AX8) 

$(\alpha \to \gamma )\to \left(\right(\beta \to \gamma )\to ((\alpha \vee \beta )\to \gamma \left)\right)$

(EM) 

$\alpha \vee \neg \alpha$

(cf) 

$\neg \neg \alpha \to \alpha$

Inference rule:

(MP) 

$\alpha \alpha \to \beta \beta$

Remark 8.

It is worth noting that (Ax1)-(Ax8) plu MP constitutes a sound and complete Hilbert calculus for positive intuitionistic logic $IP{L}^{+}$, which is semantically characterized by the class of implicative lattices with 1 as the only designated value.

5. Sette Hyperalgebras for ${\mathbf{C}}_{\mathbf{\omega }}$

In 1969, Sette and da Costa proposed in [7] the first semantics for $C_{\omega }$ by means of $C_{\omega }$-algebras, based on the notion of $C_n$-algebras (afterwards called da Costa algebras) introduced 3 years before by da Costa in [6]. $C_{\omega }$-algebras are implicative lattices expanded with an equivalence relation which is congruential with respect to the implicative lattice operations, and with an operator ′ satisfying suitable properties in order to interpret the paraconsistent negation.

Also in 1969 (but only published in 1977), M. Fidel introduced a novel algebraic-relational non-deterministic semantics for all the calculi $C_n$ (including $C_{\omega }$), nowadays known as Fidel structures, proving for the first time the decidability of da Costa’s calculi (see [16]). In 1986, A. Loparić proposed another semantical characterization for $C_{\omega }$ by means of valuation semantics over $\{0,1\}$, also known as bivaluation semantics, proving soundness and completeness (see [17]). In the same year, M. Baaz introduced in [18] a sound and complete Kripke-style Semantics for $C_{\omega }$.

In Chapter 2 of his MSc dissertation from 1971 under supervision of da Costa ([8]), Sette introduced a class of hyperalgebras for $C_{\omega }$ called $C_{\omega }$-hyperlattices. He proved that they correspond with $C_{\omega }$ algebras, inducing therefore a suitable semantics for $C_{\omega }$.

In what follows, a slightly more general definition of Sette’s hyperalgebras will be considered, giving in Theorem 1, for the first time, a direct proof of soundness and completeness of ${\mathcal{C}}_{\omega }$ w.r.t. these hyperalgebras. Recall that $\top =\mathsf{Max}\left(L\right)$, thus $w\in \top$ if and only if $z⪯w$ for every $z\in L$.

Definition 16 (Sette hyperalgebras for $C_{\omega }$).

A Sette hyperalgebra for $C_{\omega }$ (or a SHC_ω) is a hyperalgebra $\mathsf{H}=\langle H,⋏,⋎,⊸,÷\rangle$ over ${\Sigma }_{\omega }$ such that the reduct $\langle H,⋏,⋎,⊸\rangle$ is a SIHL and the hyperoperator ÷ satisfies the following properties, for every $x,y,w\in H$:

(H1) 

$y\in ÷x$ and $w\in x⋎y$ implies that $w\in \top$;

(H2) 

$y\in ÷x$ and $w\in ÷y$ implies that $w⪯x$.

By Remark 6 it is immediate to see that conditions (H1) and (H2) can be written in a concise way as follows:

(H1’) 

$x⋎÷x\equiv \top$;

(H2’) 

$÷÷x⪯x$,

for every $x\in H$.

Definition 17 (SHC_ω semantics).

Let $\mathsf{H}$ be a ${\mathrm{SHC}}_{\omega }$, and let $\Gamma \cup \left\{\phi \right\}$ be a set of formulas over ${\Sigma }_{\omega }$.

1. 

The Nmatrix associated to $\mathsf{H}$ is ${\mathcal{M}}_{\mathsf{H}}=\langle \mathsf{H},\top \rangle$.

2. 

We say that φ is a semantical consequence of Γ w.r.t. $\mathsf{H}$, denoted by $\Gamma {\vDash }_{\mathsf{H}}\phi$, if $\Gamma {\vDash }_{{\mathcal{M}}_{\mathsf{H}}}\phi$.

3. 

Let ${\mathbb{SHC}}_{\omega }$ be the class of ${\mathrm{SHC}}_{\omega }$s. Then, φ is a semantical consequence of Γ w.r.t. ${\mathrm{SHC}}_{\omega }$s, denoted by $\Gamma {\vDash }_{{\mathbb{SHC}}_{\omega }}\phi$, if $\Gamma {\vDash }_{\mathsf{H}}\phi$ for every $\mathsf{H}\in {\mathbb{SHC}}_{\omega }$.

By using the notion of swap structures, in Section 6 it will we obtained, for the first time, a direct proof of soundness and completeness of $C_{\omega }$ w.r.t. Sette hyperalgebras semantics.

Remark 9.

It is worth noting that the original formulation of hyperalgebras for $C_{\omega }$ given in [8, Definition 2.3] considered, besides (H2’), condition (H1”)$x⋎÷x=\top$. The latter condition is clearly stronger than (H1’). By virtue of Definition 17, in order to validate (EM) w.r.t. ${\mathbb{SHC}}_{\omega }$ it suffices to require the weaker condition (H1’), as we did.

6. Swap Structures for ${\mathbf{C}}_{\mathbf{\omega }}$

With the aim of obtaining a more elucidative (non-deterministic) semantics for the paraconsistent logics known as logics of formal inconsistency (LFIs), in [19, Chapter 6] was introduced a particular way to define non-deterministic matrices (or Nmatrices) called swap structures. This particular class of Nmatrices can be seen as non-deterministic twist structures (which, in turn, constitute a particular class of logical matrices), see [9]. Since the LFIs studied in [19] are based on classical logic, which is characterized by the two-elements Boolean algebra $\mathbf{2}:=\{0,1\}$, the original swap structures were defined over 2. In this section, swap structures for $C_{\omega }$ will be introduced, showing that they form a particular class of ${\mathrm{SHC}}_{\omega }$s which characterize $C_{\omega }$. As one could expect, they are defined over implicative lattices, the algebraic models for $IP{L}^{+}$.

Recall that, given an implicative lattice $\mathcal{A}=\langle A,\wedge ,\vee ,\to \rangle$ and $a\in A$, then $a\to a$ is the top element of A, which will be denoted by 1. From now on, given $z\in A\times A$, the first and second components of z will be denoted, respectively, by $z_1$ and $z_2$. That is, $z=(z_1,z_2)$.

Definition 18 (Swap structures for $C_{\omega }$).

Let $\mathcal{A}=\langle A,\wedge ,\vee ,\to \rangle$ be an implicative lattice. Let $S_{\mathcal{A}}=\{z\in A\times A:z_1\vee z_2=1\}$. The swap structure for $C_{\omega }$ over $\mathcal{A}$ is the hyperalgebra $\mathsf{S}\left(\mathcal{A}\right)=\langle S_{\mathcal{A}},\breve{\wedge },\breve{\vee },\breve{\to },\breve{\neg }\rangle$ over the signature ${\Sigma }_{\omega }$ such that the hyperoperators are defined as follows:

$\begin{array}{ccc}z\breve{\wedge }w=\{u\in S_{\mathcal{A}}:u_1=z_1\wedge w_1\}\hfill & \hfill & z\breve{\to }w=\{u\in S_{\mathcal{A}}:u_1=z_1\to w_1\}\hfill \\ z\breve{\vee }w=\{u\in S_{\mathcal{A}}:u_1=z_1\vee w_1\}\hfill & \hfill & \breve{\neg }z=\{u\in S_{\mathcal{A}}:u_1=z_{2}and{u}_2\le z_1\}\hfill \end{array}$

The hyperoperations in $S_{\mathcal{A}}$ can be described in a more succinct way as follows:

$\begin{array}{ccc}z\breve{\wedge }w=(z_1\wedge w_1,\_)\hfill & \hfill & z\breve{\to }w=(z_1\to w_1,\_)\hfill \\ z\breve{\vee }w=(z_1\vee w_1,\_)\hfill & \hfill & \breve{\neg }z=(z_2,\_\le z_1)\hfill \end{array}$

Following the usual definitions for swap structures (see for instance [19, Chapter 6] and [20]), it is possible to associate an Nmatrix to each swap structure in a natural way:

Definition 19.

Let $\mathcal{A}$ be an implicative lattice. The Nmatrix associated to $S_{\mathcal{A}}$ is $\mathcal{M}\left(\mathcal{A}\right)=\langle S_{\mathcal{A}},D_{\mathcal{A}}\rangle$ where the set of designated truth-values is $D_{\mathcal{A}}=\{z\in S_{\mathcal{A}}:z_1=1\}$.

Proposition 18.

Let $\mathsf{S}\left(\mathcal{A}\right)$ be the swap structure for $C_{\omega }$ over an implicative lattice $\mathcal{A}$. Then:

1. 

$\mathsf{S}\left(\mathcal{A}\right)$ is a ${\mathrm{SHC}}_{\omega }$ which satisfies, for every $z\in S_{\mathcal{A}}$, condition (H1”): $z⋎÷z=\top$.

2. 

The preorder in $\mathsf{S}\left(\mathcal{A}\right)$ is given as follows: $z⪯w$ iff $z_1\le w_1$ in $\mathcal{A}$. Hence, $z\equiv w$ iff $z_1=w_1$. Moreover, $D_{\mathcal{A}}=\top$.

3. 

$\mathcal{M}\left(\mathcal{A}\right)={\mathcal{M}}_{\mathsf{S}\left(\mathcal{A}\right)}$.

Proof. 

It is immediate from the definitions and the properties of implicative lattices. Item (2) uses Remark 2, specifically: $z⪯w$ iff $z\in z⋏w$. □

Definition 20 (Swap structures semantics for $C_{\omega }$).

Let $\Gamma \cup \left\{\phi \right\}$ be a set of formulas over ${\Sigma }_{\omega }$. Then, φ is a semantical consequence of Γ w.r.t. swap structures, denoted by $\Gamma {\vDash }_{C_{\omega }}^{SW}\phi$, whenever $\Gamma {\vDash }_{\mathcal{M}\left(\mathcal{A}\right)}\phi$ for every implicative lattice $\mathcal{A}$.

In order to prove our main result (Theorem 1 below), we recall here some well-known notions and results concerning (Tarskian) logics.

Given a Tarskian and finitary logic L and a set of formulas $\Delta \cup \left\{\phi \right\}$ of L, the set $\Delta$ is said to be φ-saturated in L if the following holds: (i) $\Delta {⊬}_{\mathbf{L}}\phi$; and (ii) if $\psi \notin \Delta$ then $\Delta ,\psi {⊢}_{\mathbf{L}}\phi$.

It is immediate to see that any $\phi$-saturated in a Tarskian logic is deductively closed, i.e.: $\psi \in \Delta$ iff $\Delta {⊢}_{\mathbf{L}}\psi$.

By a classical result proven by Lindenbaum and Łoś, if $\Gamma \cup \left\{\phi \right\}$ is a set of formulas of a Tarskian and finitary logic L such that $\Gamma {⊬}_{\mathbf{L}}\phi$, then there exists a $\phi$-saturated set $\Delta$ such that $\Gamma \subseteq \Delta$.2 Since $C_{\omega }$ is a Tarskian and finitary logic, Lindenbaum-Łoś theorem holds for it. We arrive to our main result:

Theorem 1 (Soundness and completeness of $C_{\omega }$ w.r.t. hypertructures).

Let $\Gamma \cup \left\{\phi \right\}$ be a set of formulas over ${\Sigma }_{\omega }$. The following assertions are equivalent:

1. 

$\Gamma {⊢}_{C_{\omega }}\phi$;

2. 

$\Gamma {\vDash }_{{\mathbb{SHC}}_{\omega }}\phi$;

3. 

$\Gamma {\vDash }_{C_{\omega }}^{SW}\phi$.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$ (Soundness of $C_{\omega }$ w.r.t. ${\mathrm{SHC}}_{\omega }$s). Assume that $\Gamma {⊢}_{C_{\omega }}\phi$. In order to prove that $\Gamma {\vDash }_{{\mathbb{SHC}}_{\omega }}\phi$, it is enough to prove the following, for every $\mathsf{H}\in {\mathbb{SHC}}_{\omega }$ and every valuation $v:For\left({\Sigma }_{\omega }\right)\to H$ over ${\mathcal{M}}_{\mathsf{H}}$: (i) if $\phi$ is an instance of an axiom of $C_{\omega }$ then $v\left(\phi \right)\in \top$; and (ii) if $v\left(\phi \right)\in \top$ and $v(\phi \to \psi )\in \top$ then $v\left(\psi \right)\in \top$. So, let $\mathsf{H}$ and v.

Axiom (Ax1): Let $\alpha =\phi \to (\psi \to \phi )$ be an instance of (Ax1), and let $x=v\left(\phi \right)$ and $y=v\left(\psi \right)$. Then, $v\left(\alpha \right)\in x⊸(y⊸x)=\top$ (see Proposition 12). Using a similar argument combined with Propositions 12, 14 and 15 we prove that if $\phi$ is an instance of the other axioms AX2-AX8 then $v\left(\phi \right)\in \top$.

Axiom (EM): Let $\alpha =\phi \vee \neg \phi$ be an instance of (EM), and let $x=v\left(\phi \right)$, $y=v(\neg \phi )$. Then, $y\in ÷x$ and so $v\left(\alpha \right)\in x⋎y\subseteq x⋎÷x=\top$, by (H1’). This means that $v\left(\alpha \right)\in \top$.

Axiom (cf): Let $\alpha =\neg \neg \phi \to \phi$ be an instance of (cf), and let $x=v\left(\phi \right)$, $y=v(\neg \phi )$, $z=v(\neg \neg \phi )$. Then, $y\in ÷x$, $z\in ÷y$ and so $z\le x$, by (H2). By Proposition 12 (2), $z⊸x=\top$. But then $v\left(\alpha \right)\in z⊸x=\top$, that is, $v\left(\alpha \right)\in \top$.

Finally, in order to prove that trueness is preserved by MP, let $x=v\left(\alpha \right)$, $y=v\left(\beta \right)$, $z=v(\alpha \to \beta )$, and suppose that $x\in \top$ and $z\in \top$. Since $z\in x⊸y$ then $x⋏z⪯y$, by (I1). Now, if $w\in H$ then $w⪯x$ and $w⪯z$ (since $x,z\in \top$) and so $w⪯x⋏z$, by definition of ⋏. From this, and the fact that $x⋏z\ne \varnothing$, it follows that $w⪯y$. Therefore, $y\in \top$.

$\left(2\right)\Rightarrow \left(3\right)$. It is immediate, by Proposition 18, items (1) and (3).

$\left(3\right)\Rightarrow \left(1\right)$ (Completeness of $C_{\omega }$ w.r.t. swap structures semantics). Suppose that $\Gamma {⊬}_{C_{\omega }}\phi$. Then, by Lindenbaum-Łoś result mentioned above, there exists a $\phi$-saturated set $\Delta$ in $C_{\omega }$ such that $\Gamma \subseteq \Delta$. Now, define a relation ${\simeq }_{\Delta }$ over $For\left({\Sigma }_{\omega }\right)$ as follows: $\alpha {\simeq }_{\Delta }\beta$ iff $\Delta {⊢}_{C_{\omega }}\alpha \to \beta$ and $\Delta {⊢}_{C_{\omega }}\beta \to \alpha$. Since $C_{\omega }$ contains positive intuitionistic logic (recall Remark 8), it follows that ${\simeq }_{\Delta }$ is an equivalence relation. Moreover, it is a congruence w.r.t. the signature ${\Sigma }_{\to }=\{\wedge ,\vee ,\to \}$. Thus, if $A_{\Delta }=For\left({\Sigma }_{\omega }\right){/}_{{\simeq }_{\Delta }}$ then the following operations over $A_{\Delta }$ are well-defined:

$${\left[\alpha \right]}_{\Delta }\wedge {\left[\beta \right]}_{\Delta }:={[\alpha \wedge \beta ]}_{\Delta },{\left[\alpha \right]}_{\Delta }\vee {\left[\beta \right]}_{\Delta }:={[\alpha \vee \beta ]}_{\Delta },\mathrm{and}{\left[\alpha \right]}_{\Delta }\to {\left[\beta \right]}_{\Delta }:={[\alpha \to \beta ]}_{\Delta }$$

where ${\left[\alpha \right]}_{\Delta }$ denotes the equivalence class of $\alpha$ w.r.t. ${\simeq }_{\Delta }$. Moreover, ${\mathcal{A}}_{\Delta }=\langle A_{\Delta },\wedge ,\vee ,\to \rangle$ is an implicative lattice, therefore $1={[\alpha \to \alpha ]}_{\Delta }$ is the top element, for every $\alpha$. Let $\mathsf{S}\left({\mathcal{A}}_{\Delta }\right)$ be the swap structure for $C_{\omega }$ over ${\mathcal{A}}_{\Delta }$ with domain $S_{{\mathcal{A}}_{\Delta }}$ as given in Definition 18. Let $v_{\Delta }:For\left({\Sigma }_{\omega }\right)\to S_{{\mathcal{A}}_{\Delta }}$ be the canonical map given by $v_{\Delta }\left(\alpha \right)=({\left[\alpha \right]}_{\Delta },{[\neg \alpha ]}_{\Delta })$ for every $\alpha$. Observe that ${\left[\alpha \right]}_{\Delta }\vee {[\neg \alpha ]}_{\Delta }={[\alpha \vee \neg \alpha ]}_{\Delta }=1$ then $v_{\Delta }$ is a well-defined map. Clearly it is a valuation over $\mathcal{M}\left({\mathcal{A}}_{\Delta }\right)$ such that $v_{\Delta }\left(\alpha \right)\in \top$ iff ${\left[\alpha \right]}_{\Delta }=v_{\Delta }{\left(\alpha \right)}_1=1$ iff $\Delta {⊢}_{C_{\omega }}\alpha$. From this, $v_{\Delta }\left(\alpha \right)\in \top$ for every $\alpha \in \Gamma$, while $v_{\Delta }\left(\phi \right)\notin \top$, given that $\Delta {⊬}_{C_{\omega }}\phi$. This shows that $\Gamma {⊭}_{\mathcal{M}\left({\mathcal{A}}_{\Delta }\right)}\phi$ and so $\Gamma {⊭}_{C_{\omega }}^{SW}\phi$.

This completes the proof. □

7. Conclusions and Final Remarks

In this paper we describe the almost unknown concepts of Morgado hyperlattices and Sette implicative hyperlattices and hyperalgebras for $C_{\omega }$, proving new properties about them. In particular, using the notion of swap structures, we obtain a new and direct proof of soundness and completeness of da Costa logic $C_{\omega }$ w.r.t. hyperalgebraic semantics based on Sette hyperalgebras for $C_{\omega }$.

In [22], R. Cignoli improved a construction of Kalman from 1958, obtaining an adjunction between the category and bounded distributive lattices and Kleene algebras by means of that he called a Kalman functor. This technique, based on the notion of twist structures, has been amply studied in the literature, and Kalman functor was adapted to other kinds of algebras (see, for instance, [23] and the references therein). As a future research, we aim to define a Kalman functor from the category of implicative m-hyperlattices to the category of Sette hyperalgebras for $C_{\omega }$, based on the notion of swap structures. Some first steps to adapt the Kalman functor to the hyperalgebraic setting by means of swap structures have been taken in [9], in the context of LFIs.

Morgado hyperlattices, as well as Sette implicative hyperlattices and hyperalgebras for $C_{\omega }$, can be an interesting ways to define, with appropriate adaptations, semantics for several non-classical logics. In addition to the applications to logic, the study and further development of hyperlattices of this kind may be of interest to the subject of hyperalgebras in general.