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

Abstract

José Morgado introduced in 1962 a novel notion of hyperlattices, which he called reticuloides. In his master’s thesis submitted in 1971 (under the supervision of Newton da Costa), Antonio M. Sette introduced a new class of implicative hyperlattices (here called SIHLs) based on Morgado’s hyperlattices. He also extended SIHLs by adding a unary hyperoperator, thus defining a class of hyperalgebras (denoted ${\mathrm{SHC}}_{\omega }$) corresponding to da Costa algebras for $C_{\omega }$, thereby providing suitable semantics for the logic $C_{\omega }$. In this paper, after providing a (hyper)lattice-theoretic characterization of Sette’s implicative hyperlattices and proving some basic results on SIHLs, we introduce a class of swap structures—special hyperalgebras over the signature of $C_{\omega }$ that arise naturally from implicative lattices. We prove that these swap structures are indeed ${\mathrm{SHC}}_{\omega }$. Finally, we demonstrate that the class ${\mathrm{SHC}}_{\omega }$, as well as the aforementioned swap structures, characterizes the logic $C_{\omega }$.

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 semantic framework which aims 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 M. 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, which appears to be original, to the best of our knowledge 1. He introduced the term “reticuloide” (reticuloid) to denote hyperlattices and coined 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 quasiordered 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 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 proposed 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 the fundamental properties of these structures, which in turn allow us to prove the soundness and completeness of $C_{\omega }$ with respect to ${\mathrm{SHC}}_{\omega }$ (Sette hyperalgebras for $C_{\omega }$) semantics. Furthermore, we introduce a natural subclass of SIHLs defined by swap structures, which also provide suitable semantics for $C_{\omega }$.

2. Morgado Reticuloids

There are several definitions of hyperalgebra in the literature, considering that each application of hyperalgebra in a specific area of Mathematics (mainly algebra and logic) requires a particular adaptation. Here, we adapt the notion of hyperalgebra used in [9].

Definition 1.

A hyperalgebraic signature is a sequence of pairwise disjoint sets

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

where ${\Sigma }_n=S_n\bigsqcup M_n$, $S_n$ is the set of strict hyperoperation symbols, and $M_n$ is the set of hyperoperation symbols. In particular, ${\Sigma }_0=S_0\bigsqcup M_0$, where $S_0$ 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

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

Definition 2.

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

1. 

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

2. 

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

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

Definition 3.

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

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

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

Remark 1.

1. 

Every algebraic signature $\Sigma ={(F_n)}_{n\in \mathbb{N}}$ 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 $\Sigma =({(S_n)}_{n\in \mathbb{N}},{(M_n)}_{n\in \mathbb{N}})$ naturally induces 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, hyperalgebras

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

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 satisfy 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 (or 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 $\mathsf{Min}(B)=\{x\in B:x⪯B\}$, and the set of maxima of B is $\mathsf{Max}(B)=\{x\in B:B⪯x\}$.

2. 

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

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 $x⋎y=\mathsf{Min}(Ub(\{x,y\}))$.

2. 

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

3. 

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

It is immediate from the definition above that in any m-hyperlattice,

$$(R_{⪯})x\in x⋏y\mathrm{iff}y\in x⋎y$$

and each of the conditions above is equivalent to $x⪯y$.

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 it 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. 

$x\in x⋏y$ iff $y\in x⋎y$, for every $x,y\in L$.

2. 

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

3. 

$x⋏y⪯x$ and $x⋏y⪯y$.

4. 

$x⪯x⋎y$ and $y⪯x⋎y$.

5. 

$x⋏y⪯x⋎y$.

Proof. 

• Assume that $x\in x⋏y$. By R4, $x\in y⋏x$, and so $y\in y⋎x$, by R6. Using R1, it follows that $y\in x⋎y$. Conversely, suppose that $y\in x⋎y$. By R3, $x\in x⋏y$.
• Since L is a hyperalgebra, there exists some $z\in x⋏x$, which implies (by R6) that $x\in x⋎z$. By R3 we get $x\in x⋏x$; hence $x⪯x$.
  Now, suppose that $x⪯y$ and $y⪯z$. Then $y\in x⋎y$ and $z\in y⋎z$, which implies (by R2) that $y⋎z=x⋎z$. In particular, $z\in x⋎z$, and so $x⪯z$.
• 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 R4 and R3, we get $x\in x⋏z=z⋏x$. By R6 we get $z\in z⋎x$, and so $z\in x⋎z$, by R1. This means that $x⪯z$. Similarly we get $y⪯z$.
• We then combine items (2)–(4). Indeed, ⪯ is a preorder, by (2). By (3) and (4), $x⋏y⪯x$, and $x⪯x⋎y$. By the transitivity of ⪯, $x⋏y⪯x⋎y$.

□

Example 1.

1. 

Every lattice is an m-hyperlattice.

2. 

([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 $\langle \mathcal{P}(V),⋎,⋏\rangle$ is an m-hyperlattice.

3. 

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

4. 

Let $\mathcal{C}$ be a small category with binary products ∏ and coproducts ∐ (see, for instance, [10]), and for $a, b\in Obj(\mathcal{C})$ (the set of objects of $\mathcal{C}$), define

$$a⪯biffthere exist a morphism f:a\to bin\mathcal{C}.$$

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

5. 

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 $\langle L,⪯\rangle$ is a proset. Moreover, the sets $\alpha ⋏\beta$ and $\alpha ⋎\beta$ are non-empty 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 $\gamma ⊢\beta$ imply that $\gamma ⊢\alpha \wedge \beta$.

(ii) 

$\alpha ⊢\alpha \vee \beta$ and $\beta ⊢\alpha \vee \beta$, and $\alpha ⊢\gamma$ and $\beta ⊢\gamma$ imply that $\alpha \vee \beta ⊢\gamma$.

Then, L is an m-hyperlattice. It can be shown that $\alpha ⋏\beta =\{\gamma :\gamma ⊢\alpha \wedge \beta and\alpha \wedge \beta ⊢\gamma \}$, and $\alpha ⋎\beta =\{\gamma :\gamma ⊢\alpha \vee \beta and\alpha \vee \beta ⊢\gamma \}$.

6. 

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

Definition 7.

The MHL category of hyperlattices is the one whose objects are hyperlattices and whose morphisms 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 $f(z)\in f(x)⋏f(y)$;

2. 

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

Proposition 1.

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

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, [11]) can be easily obtained for prosets and thus for m-hyperlattices. Indeed, if $\mathsf{P}=\langle P,⪯\rangle$ is a proset, so is its dual ${\mathsf{P}}^{*}=\langle P,⪰\rangle$, 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 an 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 the 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 ⋏) also holds in any m-hyperlattice.

Proposition 2.

Let $\mathsf{P}=\langle P,⪯,⋏,⋎\rangle$ be an 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 $y⪯z$.	4. $z⪯x$ iff $z⪯y$.
2. $x⪯B$ iff $y⪯B$.	5. $B⪯x$ iff $B⪯y$.
3. $x⋎z=y⋎z$.	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

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

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, $\varnothing \ne A,B\subseteq L$, and $x,y,z\in L$. Then,

1. 

$A\#B=B\#A$ for $\#\in \{⋏,⋎\}$.

2. 

If $A\subseteq B\subseteq L$, then $\mathsf{Max}(A)⪯\mathsf{Max}(B)$.

3. 

If $A\subseteq B\subseteq L$, then $\mathsf{Min}(B)⪯\mathsf{Min}(A)$.

4. 

If $x⪯y$, then $x⋏z⪯y⋏z$.

5. 

If $x⪯y$, then $x⋎z⪯y⋎z$.

6. 

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

7. 

If $a⪯x$ and $a⪯B$, then $a⪯x⋏B$.

8. 

If $x⪯a$ and $B⪯a$, then $x⋎B⪯a$.

9. 

If $A⪯z$ and $B⪯z$, then $A⋎B⪯z$.

10. 

If $x⋏A⪯z$, then $x⋏(A⋏B)⪯z$.

11. 

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. 

• This 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 $x\in \mathsf{Max}(A)$ and $y\in \mathsf{Max}(B)$. Since $x\in A$, $x\in B$; hence $x⪯y$. We conclude that $\mathsf{Max}(A)⪯\mathsf{Max}(B)$.
• This follows from (2) by duality.
• Assume that $x⪯y$. By definition, $x⋏z=\mathsf{Max}(A)$ and $y⋏z=\mathsf{Max}(B)$, where $A=\{w\in L:w⪯x and w⪯z\}$, and $B=\{w\in L:w⪯y and w⪯z\}$. By the definition of m-hyperlattices, $\varnothing \ne A\subseteq B\subseteq L$. By item (2), $x⋏z⪯y⋏z$.
• This follows from (4) by duality.
• 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, $a⪯x⋏B$.
• This follows from (7) by duality.
• 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, $x⋏(A⋏B)⪯z$.
• This follows from (10) and by the 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, $x⋏(y⋏z)⪯(x⋏y)⋏z$.
• This follows from (12) by duality.

□

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 (5). Let $p, q, r, s$ be four 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 ⋎ with respect to 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 (5). Indeed, take $A=\{p,q\}$ and $x=r$ for $p, q, r$ different propositional variables. Since $p\wedge r⪯/q$, $\{p,q\}⋏r⪯/\{p,q\}$. Analogously, since $q⪯/p\vee r$, $\{p,q\}⪯/\{p,q\}⋎r$. In order to guarantee the validity of these desirable properties, stability is required once again; see Proposition 4 below.

Definition 8.

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

$$A⋏B=a⋏b and A⋎B=a⋎b.$$

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

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

Since $x\in \mathrm{Lb}(\{a,b\})$, it follows that $x⪯y$. 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 $a⋏c⪯b⋏d$, providing that $A⋏C=a⋏c⪯b⋏d=B⋏D$. The monotonicity of ⋎ follows from the monotonicity of ⋏ by duality. □

Thanks to stability, supremoids and infimoids between stable sets can be characterized in a natural way. Moreover, it follows that both hyperoperators are associative:

Proposition 6.

Let $A,B\subseteq L$ be stable sets. Then,

1. 

$A⋏B=\mathsf{Max}(Lb(A\cup B))$, and $A⋎B=\mathsf{Min}(Ub(A\cup B))$.

2. 

Let $\#\in \{⋏,⋎\}$. If $x\in A\#B$ and $y\equiv x$, then $y\in A\#B$.

3. 

$(x⋏y)⋏z=x⋏(y⋏z)$.

4. 

$(x⋎y)⋎z=x⋎(y⋎z)$.

Proof. 

• Let $a\in A$ and $b\in B$. By the stability of both A and B, it is immediate that $Lb(A\cup B)=Lb(\{a,b\})$. From this and by Proposition 4, $A⋏B=a⋏b=\mathsf{Max}(Lb(\{a,b\}))=\mathsf{Max}(Lb(A\cup B))$. The case for $A⋎B$ follows from the case for ⋏ by duality.
• Let $x\in A⋏B$. By item (1), $A⋏B=\mathsf{Max}(Lb(A\cup B))$. Suppose that $y\equiv x$, and let $z\in A\cup B$. Then, $y⪯x⪯z$, and so $y⪯z$. That is, $y\in Lb(A\cup B)$. Let $w\in Lb(A\cup B)$. Hence, $w⪯x⪯y$, and so $w⪯y$. This means that $y\in \mathsf{Max}(Lb(A\cup B))=A⋏B$. The case for ⋎ follows from here by duality.
• By item (12) of Proposition 3, $(x⋏y)⋏z\equiv x⋏(y⋏z)$. In turn, $x⋏(y⋏z)=A⋏B$ where $A=\left\{x\right\}$ and $B=y⋏z$ are stable. Let $w\in (x⋏y)⋏z$ and $a\in A⋏B$. Then, $w\equiv a$, and so $w\in A⋏B$, by item (2). This shows that $(x⋏y)⋏z\subseteq x⋏(y⋏z)$. The converse inclusion is proved analogously.
• This follows from item (3) by duality.

□

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

Proposition 7

(Absorption laws for m-hyperlattices). Let L be an m-hyperlattice, and let $x,y\in L$. Then,

$$x\in x⋏(x⋎y)=x⋎(x⋏y).$$

Proof. 

Let $A=x⋏(x⋎y)$. Since $\left\{x\right\}$ and $x⋎y$ are stable, $A=\mathsf{Max}(Lb(\left\{x\right\}\cup (x⋎y))))$, by Proposition 6 (1). Since $x⪯x$ and $x⪯x⋎y$, it follows that $x⪯A$. Conversely, let $a\in A$. Then $a⪯x$, since $a\in Lb(\{x\}\cup (x⋎y))$. This shows that $A⪯x$, and so $A\equiv x$. Now, let $z\in A$. Hence, $x\equiv z$. Taking into account that $A=\left\{x\right\}⋏(x⋎y)$ where $\left\{x\right\}$ and $x⋎y$ are stable, it follows by Proposition 6 (2) that $x\in x⋏(x⋎y)$. Finally, let $B=x⋎(x⋏y)$. By a similar argument to the one given in the proof of item (3) of Proposition 6, it is easy to show that $A=B$. □

Remark 6.

It is worth mentioning at this point that, in [12], X. Guo and X. Xin proposed a notion of hyperlattices which is closely related to Morgado’s one. For them, a hyperlattice is a hyperstructure $\langle L,\otimes ,\oplus \rangle$ such that the binary hyperoperators $\otimes$ (infimum) and ⊕ (supremum) satisfy the following properties, for every $x,y,z\in L$ and $\#\in \{\otimes ,\oplus \}$:

(1) 

$x\in x\#x;$

(2) 

$x\#y=y\#x;$

(3) 

$(x\#y)\#z=x\#(y\#z);$

(4) 

$x\in x\otimes (x\oplus y)andx\in x\oplus (x\otimes y).$

By the results already proven in this section, it follows that every m-hyperlattice is a hyperlattice in the sense of [12]. However, the converse is not true. Indeed, it can be proven that the 3-element hyperlattice proposed in ([13] Example 26) satisfies the definition of the hyperlattice of [12]. However, it contains elements $c, a$ such that $c\in c\otimes a$, but $a\notin c\oplus a$. Hence, condition $(R_{⪯})$ of m-hyperlattices (stated right below Definition 6) is not valid, in general, in this class of hyperlattices, and so no preorder is available. From this, we can see that both notions of hyperlattices are different.

Proposition 8.

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

$$A⋏(B⋏C)=(A⋏B)⋏C and A⋎(B⋎C)=(A⋎B)⋎C.$$

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 an m-hyperlattice. The sets $\mathsf{Min}(P)$ and $\mathsf{Max}(P)$ of the minimum and maximum elements of P will be denoted by ⊥ (or 0) and ⊤ (or 1), respectively. We say that an m-hyperlattice P is bounded if $\top \ne \varnothing \ne \perp$.

Remark 7.

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 an m-hyperlattice. We define M-terms recursively as follows:

1. 

Every stable subset $A\subseteq L$ is an 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 an 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] (which we refer to as Sette implicative hyperlattices) as a basis for the hyperalgebraic semantics for da Costa logic $C_{\omega }$ which 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 developed in [8] were pointing in the right direction.

Definition 11

(Sette implicative hyperlattices, [8] (Definition 2.3)). A Sette implicative hyperlattice (or an SIHL) is a hyperalgebra $\mathsf{L}=\langle L,⋏,⋎,⊸\rangle$ such that the reduct $\langle L,⋏,⋎\rangle$ is an 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 $z\in x⊸y$ imply that $z^{\prime }\in x⊸y$.

We can provide a more direct characterization of SIHLs.

Definition 12.

Let $\mathsf{P}$ be an 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 an m-hyperlattice. Then, $\mathsf{L}$ is an SIHL iff $x⊸y=\mathsf{Max}(\mathsf{R}(x,y))$, 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, $z^{\prime }⪯z$. This proves that $\mathsf{R}(x,y)⪯z$, and so $z\in \mathsf{Max}(\mathsf{R}(x,y))$. That is, $x⊸y\subseteq \mathsf{Max}(\mathsf{R}(x,y))$. In order to prove the converse inclusion, let $z\in \mathsf{Max}(\mathsf{R}(x,y))$. 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 $z^{\prime }\in x⊸y$ (note that $x⊸y\ne \varnothing$, since $\mathsf{L}$ 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, $\mathsf{Max}(\mathsf{R}(x,y))\subseteq x⊸y$.

• If’ part: Assume that $x⊸y=\mathsf{Max}(\mathsf{R}(x,y))$, for every $x,y\in L$. Since $\mathsf{L}$ is a hyperalgebra, by hypothesis, $x⊸y\ne \varnothing$, for every $x,y\in L$. If $z\in x⊸y$, then, by the 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 $z⪯\mathsf{Max}(\mathsf{R}(x,y))=x⊸y$. 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 $z^{\prime }\in \mathsf{Max}(\mathsf{R}(x,y))=x⊸y$. That is, ⊸ also satisfies (I3). □

Remark 8.

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

Corollary 1.

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

Lemma 2.

Let $\mathsf{L}$ be an SIHL. 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.

Proof. 

• This follows from $x⋏y⪯y$.
• This follows from $y\in R(x,y)$ and $x⊸y=\mathrm{Max}(R(x,y))$.
• If $z\in R(x,y)$, then $x⋏z⪯y⪯y^{\prime }$, which means that $z\in R(x,y^{\prime })$. 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 $z\in R(x^{\prime },y)$ implies $z\in R(x,y)$. 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

  $$R(x,y)=R(x^{\prime },y)=R(x^{\prime },y^{\prime }).$$
• This follows from $x⋏(z⋏z^{\prime })⪯x⋏z$ and $x⋏(z⋏z^{\prime })⪯x⋏z^{\prime }$.
• This 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 $z\in \mathrm{Ub}(R(x,y))$ and $z^{\prime }\in x⊸y=\mathrm{Max}(\mathrm{Ub}(R(x,y)))$, we get $z^{\prime }⪯z$. Similarly $z⪯z^{\prime }$.

□

In the remainder of this section, some basic but useful properties of SIHLs will be obtained. They will be used in Section 6.

Proposition 11.

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

1. 

$x⋏(x⊸y)⪯y$.

2. 

$x⪯y$ if and only if $x⊸y=\top$.

3. 

$x⊸x=\top$.

4. 

$x⊸(y⊸x)=\top$.

5. 

$x⊸(y⊸(x⋏y))=\top$.

6. 

$(x⋏y)⊸x=\top$, and $(x⋏y)⊸y=\top$.

7. 

$x⊸(x⋎y)=\top$, and $y⊸(x⋎y)=\top$.

Proof. 

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

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

□

Proposition 12.

Let $A,B\subseteq L$ be stable, and let $a\in A$ and $b\in B$.

1. 

$A⊸B=a⊸b$; hence $A⊸B$ is stable.

2. 

Let $R(A,B)=\{z\in L:A⋏z⪯B\}$. Then $R(A,B)=R(a,b)$, and so $A⊸B=\mathsf{Max}(\mathsf{R}(A,B))$.

3. 

If $x\in A⊸B$ and $y\equiv x$, then $y\in A⊸B$.

Proof. 

• This follows from Lemma 2 items (5) and (9).
• Let $z\in R(A,B)$. Then, $A⋏z⪯B$. In particular, $a⋏z⪯b$, and so $z\in R(A,b)$. Conversely, let $z\in R(a,b)$, and let $x\in A$ and $y\in B$. Since A and B are stable, $x⋏z⪯a⋏z⪯b⪯y$. Then, $A⋏z⪯B$, and so $z\in R(A,B)$. This shows that $R(A,B)=R(a,b)$. From this, using item (1), $A⊸B=a⊸b=\mathsf{Max}(\mathsf{R}(a,b))=\mathsf{Max}(\mathsf{R}(A,B))$.
• This is an immediate consequence of item (1) and property (I3) of SIHLs.

□

Proposition 13.

Let $\mathsf{L}$ be an 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, if $A,B$ are stable, then $A⪯B$ iff $a⊸b=\top$ for some $a\in A$ and some $b\in B$ iff $a⪯b$ 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. 

$A⋏(A⊸B)⪯B$.

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 11 (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 $w\in y⊸z=\mathsf{Max}(\mathsf{R}(y,z))$. 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 the definition of SIHLs.
• Assume that $A⪯B⊸C$, and let $a\in A$, $b\in B$, and $c\in C$. Since $b⊸c\subseteq B⊸C$, $a⪯b⊸c$, 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 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.

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

$$x⋏(y⋎z)=(x⋏y)⋎(x⋏z)andx⋎(y⋏z)=(x⋎y)⋏(x⋎z).$$

The fact that SIHLs are distributive hyperlattices will allow us to prove, in Section 6, the soundness of some important axioms of positive intuitionistic logic with respect to implicative hyperlattices (see the proof of Theorem 1, $(1)\Rightarrow (2)$).

Proposition 14.

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

1. 

$x⋏(y⋎z)\equiv (x⋏y)⋎(x⋏z)$.

2. 

$(x⊸(y⊸z))⊸((x⊸y)⊸(x⊸z))=\top$.

3. 

$(x⊸y)⊸((y⊸z)⊸(x⋎y)⊸z))=\top$.

4. 

$(x⊸y)⋏(y⊸z)⪯(x⋎y)⊸z)$.

5. 

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

6. 

$x⋎(y⋏z)\equiv (x⋎y)⋏(x⋎z)$.

Proof. 

• Recall Proposition 6, and let

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

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

  $$w\in (x⋏y)⋎(x⋏z)=\mathrm{Min}(\mathrm{Ub}((x⋏y)\cup (x⋏z))).$$
  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 (2) of Proposition 13 that $x⋏u⪯w$. By Proposition 4, this implies that $x⋏(y⋎z)=x⋏u⪯w$. Therefore, $x⋏(y⋎z)⪯(x⋏y)⋎(x⋏z)$.
• By Proposition 13 items (1) and (3), the following holds:

  $$\begin{array}{cc}\hfill & (x⊸(y⊸z))⊸((x⊸y)⊸(x⊸z))=\top \mathrm{iff}\hfill \\ \hfill & x⊸(y⊸z)⪯(x⊸y)⊸(x⊸z) \mathrm{iff}\hfill \\ \hfill & (x⊸(y⊸z))⋏(x⊸y)⪯x⊸z \mathrm{iff}\hfill \\ \hfill & [(x⊸(y⊸z))⋏(x⊸y)]⋏x⪯z\hfill \end{array}$$
  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 11 (1), and $y⋏(x⋏A)⪯z$. Indeed, $x⋏A⪯y⊸z$, by Proposition 13 (4); hence $y⋏(x⋏A)⪯z$, by Proposition 13 (3).
  Since $B⋏x⪯y$ and $y⋏(x⋏A)⪯z$, we get, by Proposition 5, that $(B⋏x)⋏(x⋏A)⪯y⋏(x⋏A)⪯z$. 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 13 items (1) and (3), the following holds:

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

  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), it is enough to prove that $A⋏x⪯z$ and $A⋏y⪯z$.
  Since $A⋏y\equiv (x⊸y)⋏[y⋏(y⊸z)]$ and $y⋏(y⊸z)⪯z$, we get $A⋏y⪯z$. For the other part, note that $A⋏x\equiv [x⋏(x⊸y)]⋏(y⊸z)$, and then

  $$A⋏x\equiv [x⋏(x⊸y)]⋏(y⊸z)⪯y⋏(y⊸z)⪯z.$$
• This is an immediate consequence of the previous item (3) and items (1) and (3) of Proposition 13.
• 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 13 (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. 

This follows from items (1) and (6) of Proposition 14, by applying an argument similar to the one given in the proof of item (3) of Proposition 6. □

Proposition 15.

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

$$A⋏(B⋎C)=(A⋏B)⋎(A⋏C) and A⋎(B⋏C)=(A⋎B)⋏(A⋎C).$$

Proof. 

By an argument similar to that 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 Corollary 2 we have $a⋏(b⋎c)=(a⋏b)⋎(a⋏c)$, we get $A⋏(B⋎C)=(A⋏B)⋎(A⋏C)$. The proof for the other equation is similar. □

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 an SIHL. We define S-terms recursively as follows:

1. 

Every stable 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 12. □

Proposition 17.

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

Proof. 

By stability, we only need to prove that $b⊸c⪯a⊸d$ for some $a\in A$, $b\in B$, $c\in C$, and $d\in D$. By Proposition 13 (3), the latter is equivalent to prove that $(b⊸c)⋏a⪯d$. By hypothesis, $a⪯b$ and $c⪯d$. By stability and by Propositions 5 and 11 (1), $(b⊸c)⋏a⪯(b⊸c)⋏b⪯c⪯d$. This concludes the proof. □

4. The Logic ${\mathit{C}}_{\mathsf{\omega }}$

Among the most influential contributions 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 [14]. The logic $C_{\omega }$ and the other systems $C_n$ are defined over the signature ${\Sigma }_{\omega }=\{\wedge ,\vee ,\to ,\neg \}$. Da Costa’s original idea 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 [15], $C_{\omega }$ is not the deductive limit of these calculi. Despite this, $C_{\omega }$ has several interesting features: it is based on positive intuitionistic logic, unlike the logics $C_n$ (for $n<\omega$), which rely on positive classical logic. Indeed, while the latter satisfy Peirce’s law $((\alpha \to \beta )\to \alpha )\to \alpha$, the former do not (see ([16] Theorem 15)). On the other hand, each $C_n$ (for $n<\omega$) is finitely trivializable, while $C_{\omega }$ is not (see ([16] 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: 

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

AX3: 

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

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: 

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

EM: 

$\alpha \vee \neg \alpha$

cf: 

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

• Inference rule: 

MP: 

$\frac{\alpha \alpha \to \beta }{\beta }$

Remark 9.

It is worth noting that AX1–AX8 plus MP constitutes a sound and complete Hilbert calculus for positive intuitionistic logic ${\mathit{IPL}}^{+}$, which is semantically characterized by the class of implicative lattices with 1 as the only designated value.

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

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 three 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 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 [17]). 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 [18]). In the same year, M. Baaz introduced in [19] sound and complete Kripke-style semantics for $C_{\omega }$.

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

In what follows, a slightly more general definition of Sette’s hyperalgebras will be considered, giving a direct proof of the soundness and completeness of ${\mathcal{C}}_{\omega }$ with respect to these hyperalgebras in Theorem 1. Recall that $\top =\mathsf{Max}(L)$; 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 ${\mathit{SHC}}_{\omega }$) is a hyperalgebra $\mathsf{H}=\langle H,⋏,⋎,⊸,÷\rangle$ over ${\Sigma }_{\omega }$ such that the reduct $\langle H,⋏,⋎,⊸\rangle$ is an SIHL and the hyperoperator ÷ satisfies the following properties, for every $x,y,w\in H$:

H1: 

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

H2: 

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

By Remark 7 it is immediate 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

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

1. 

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

2. 

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

3. 

Let ${\mathbb{SHC}}_{\omega }$ be the class of ${\mathit{SHC}}_{\omega }$. Then, φ is a semantical consequence of Γ with respect to ${\mathit{SHC}}_{\omega }$, 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, for the first time, a direct proof will be obtained of the soundness and completeness of $C_{\omega }$ with respect to Sette hyperalgebra semantics.

Remark 10.

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 with respect to ${\mathbb{SHC}}_{\omega }$, it suffices to require the weaker condition H1’, as adopted above.

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

With the aim of obtaining more elucidative (non-deterministic) semantics for the paraconsistent logics known as logics of formal inconsistency (LFIs), in [20] Chapter 6, a particular way to define non-deterministic matrices (or Nmatrices) referred to as swap structures was introduced. This particular class of Nmatrices can be seen as non-deterministic twist structures (which, in turn, constitute a class of logical matrices); see [9]. In [21], swap structures were also introduced to deal with some non-normal, non-self-extensional modal logics sometimes referred to as Ivlev-like modal logics. Since the logics studied in [20,21] are based on classical logic, which is characterized by the two-element Boolean algebra $\mathbf{2}:=\{0,1\}$, the swap structures considered in these papers were defined over $\mathbf{2}$. In this section, swap structures for $C_{\omega }$ will be introduced, showing that they form a particular class of ${\mathrm{SHC}}_{\omega }$, which characterize $C_{\omega }$. As might be expected, given that implicative lattices are the algebraic models for ${\mathrm{IPL}}^{+}$, the swap structures for $C_{\omega }$ are defined over these lattices rather than over $\mathbf{2}$ (and over Boolean algebras in general). This is the first example of swap structures outside of the context of LFIs and Ivlev-like modal logics.

Recall that, given an implicative lattice $\mathcal{A}=\langle A,\wedge ,\vee ,\to \rangle$ and $a\in A$, $a\to a$ 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 $\mathsf{S}(\mathcal{A})=\langle S_{\mathcal{A}},\breve{\wedge },\breve{\vee },\breve{\to },\breve{\neg }\rangle$ 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 more succinctly 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 standard definitions for swap structures (see, for instance, [20] Chapter 6 and [21]), 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 with $\mathsf{S}(\mathcal{A})$ is $\mathcal{M}(\mathcal{A})=\langle \mathsf{S}(\mathcal{A}),D_{\mathcal{A}}\rangle$ where the set of designated truth-values is $D_{\mathcal{A}}=\{z\in S_{\mathcal{A}}:z_1=1\}$.

Proposition 18.

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

1. 

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

2. 

The preorder in $\mathsf{S}(\mathcal{A})$ 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. 

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

Proof. 

This 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 structure 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 with respect to swap structures, denoted by $\Gamma {\vDash }_{C_{\omega }}^{SW}\phi$, whenever $\Gamma {\vDash }_{\mathcal{M}(\mathcal{A})}\phi$ for every implicative lattice $\mathcal{A}$.

Example 2.

Recall the 3-element Heyting algebra defined over the chain $0\le {\displaystyle \frac{1}{2}}\le 1$ with the following truth-tables:where $\sim x=x\to 0$. The ∼-less reduct of the above Heyting algebra is a 3-valued implicative lattice, which we will call ${\mathcal{A}}_3$. Let us analyze the swap structure $\mathsf{S}({\mathcal{A}}_3)$. Its domain is $S_{{\mathcal{A}}_3}=\{T,t,t_0,f,F\}$ such that

$$\begin{array}{ccccc}T=(1,0),\hfill & t=(1,{\displaystyle \frac{1}{2}}),\hfill & t_0=(1,1),\hfill & f=({\displaystyle \frac{1}{2}},1),\hfill & F=(0,1).\hfill \end{array}$$

According to Definition 18, the hyperoperations over $\mathsf{S}({\mathcal{A}}_3)$ are defined as follows:where $D_{{\mathcal{A}}_3}=\{T,t,t_0\}$. Observe that the preorder ⪯ in $\mathsf{S}({\mathcal{A}}_3)$ is given by $F⪯f⪯T\equiv t\equiv t_0$. In the Nmatrix $\mathcal{M}({\mathcal{A}}_3)=\langle \mathsf{S}({\mathcal{A}}_3),D_{{\mathcal{A}}_3}\rangle$ associated with this ${\mathit{SHC}}_{\omega }$, $t_0$ is the only inconsistent (or paraconsistent) value: $t_0\in D_{{\mathcal{A}}_3}$ and $\breve{\neg }{t}_0\subseteq D_{{\mathcal{A}}_3}$. In turn, f and F are such that $f\breve{\vee }(f\breve{\to }F)=\left\{f\right\}\subseteq /D_{{\mathcal{A}}_3}$, invalidating the Peirce rule $\alpha \vee (\alpha \to \beta )$ in $\mathcal{M}({\mathcal{A}}_3)$. It is easy to check that $\mathcal{M}({\mathcal{A}}_3)$ is a hyperalgebraic model of $C_{\omega }$.

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 $\phi$-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 that any $\phi$-saturated set 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 at our main result:

Theorem 1

(Soundness and completeness of $C_{\omega }$ with respect to hyperstructures). 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. 

• $(1)\Rightarrow (2)$ (Soundness of $C_{\omega }$ with respect to ${\mathrm{SHC}}_{\omega }$). 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 $v:For({\Sigma }_{\omega })\to H$ over ${\mathcal{M}}_{\mathsf{H}}$: (i) if $\phi$ is an instance of an axiom of $C_{\omega }$, then $v(\phi )\in \top$, and (ii) if $v(\phi )\in \top$ and $v(\phi \to \psi )\in \top$, then $v(\psi )\in \top$. So, let $\mathsf{H}$ and v.

Axiom AX1: Let $\alpha =\phi \to (\psi \to \phi )$ be an instance of AX1, and let $x=v(\phi )$ and $y=v(\psi )$. Then, $v(\alpha )\in x⊸(y⊸x)=\top$, by Proposition 11 (4). Using a similar argument combined with Propositions 11, 13, and 14, we prove that if $\phi$ is an instance of the other axioms AX2-AX8, then $v(\phi )\in \top$.

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

Axiom cf: Let $\alpha =\neg \neg \phi \to \phi$ be an instance of cf, and let $x=v(\phi )$, $y=v(\neg \phi )$, $z=v(\neg \neg \phi )$. Then, $y\in ÷x$, $z\in ÷y$, and so $z⪯x$, by (H2). By Proposition 11 (2), $z⊸x=\top$. But then $v(\alpha )\in z⊸x=\top$; that is, $v(\alpha )\in \top$.

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

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

$(3)\Rightarrow (1)$ (Completeness of $C_{\omega }$ with respect to swap structure semantics). Suppose that $\Gamma {⊬}_{C_{\omega }}\phi$. Then, by the 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 $For({\Sigma }_{\omega })$ 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 9), it follows that ${\simeq }_{\Delta }$ is an equivalence relation. Moreover, it is a congruence with respect to the signature ${\Sigma }_{\to }=\{\wedge ,\vee ,\to \}$. Thus, if $A_{\Delta }=For({\Sigma }_{\omega }){/}_{{\simeq }_{\Delta }}$, 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 $\alpha$ with respect to ${\simeq }_{\Delta }$. 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 $\mathsf{S}({\mathcal{A}}_{\Delta })$ be the swap structure for $C_{\omega }$ over ${\mathcal{A}}_{\Delta }$ with domain $S_{{\mathcal{A}}_{\Delta }}$, as given in Definition 18. Let $v_{\Delta }:For({\Sigma }_{\omega })\to S_{{\mathcal{A}}_{\Delta }}$ be the canonical map given by $v_{\Delta }(\alpha )=({\left[\alpha \right]}_{\Delta },{[\neg \alpha ]}_{\Delta })$ 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 $\mathcal{M}({\mathcal{A}}_{\Delta })$ such that $v_{\Delta }(\alpha )\in \top$ iff ${\left[\alpha \right]}_{\Delta }=v_{\Delta }{(\alpha )}_1=1$ iff $\Delta {⊢}_{C_{\omega }}\alpha$. From this, $v_{\Delta }(\alpha )\in \top$ for every $\alpha \in \Gamma$, while $v_{\Delta }(\phi )\notin \top$, given that $\Delta {⊬}_{C_{\omega }}\phi$. This shows that $\Gamma {⊭}_{\mathcal{M}({\mathcal{A}}_{\Delta })}\phi$, and so $\Gamma {⊭}_{C_{\omega }}^{SW}\phi$.

This completes the proof. □

7. Conclusions and Final Remarks

This paper introduces the lesser-known concepts of Morgado hyperlattices and Sette implicative hyperlattices, along with Sette hyperalgebras for $C_{\omega }$, and derives new properties of these structures. In particular, using the notion of swap structures, we obtain a new and direct proof of the soundness and completeness of da Costa logic $C_{\omega }$ with respect to hyperalgebraic semantics based on Sette hyperalgebras for $C_{\omega }$. This example confirms the fact that swap structures constitute a natural and straightforward way to find a hyperalgebraic class of models characterizing certain non-algebraizable logics. Indeed, swap structure semantics can be defined for a wide class of logic systems—whether characterized by bivaluation semantics (a useful yet non-explanatory semantic tool) or from Hilbert-style axiomatizations—in a systematic way; see, e.g., [23].

As for future work, we plan to extend the hyperalgebraic semantics based on m-hyperlattices to other non-classical logics. In particular, we aim to apply this semantic framework to several LFIs. Since most LFIs studied in the literature are based on classical logic (see, e.g., [20]), this investigation will naturally lead to the development of hyper-Boolean algebras based on m-hyperlattices. The extension of this framework to other systems within da Costa’s hierarchy $C_n$ (for $1\le n<\omega$), however, presents additional challenges. Indeed, as shown in [24], the semantic analysis of these systems requires restricted Nmatrices (RNmatrices)—that is, Nmatrices with constrained sets of admissible valuations. Because of this, our framework must be generalized to accommodate RNmatrices rather than standard Nmatrices.

Another direction for future research involves investigating categorical relationships between the classes of hyperalgebras underlying the swap construction, analogous to established results for twist structures in the context of algebraic logic. In [25], R. Cignoli improved a construction of Kalman’s from 1958, obtaining an adjunction between the category of bounded distributive lattices and the category of Kleene algebras by means of what he called a Kalman functor. This technique, based on the notion of twist structures, has been amply studied in the literature, and the Kalman functor was adapted to other kinds of algebras (see, for instance, [26] and the references therein). As for 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 initiated in [9], in the context of LFIs.

Beyond their applications to formal logic, we consider that the study and further development of Morgado hyperlattices and Sette implicative hyperlattices may also contribute to the general theory of hyperalgebras.