REMARKS ON AN ALGEBRAIC SEMANTICS FOR PARACONSISTENT NELSON ’ S LOGIC

In the paper Busaniche and Cignoli (2009) we presented a quasivariety of commutative residuated lattices, called NPc-lattices, that serves as an algebraic semantics for paraconsistent Nelson’s logic. In the present paper we show that NPc-lattices form a subvariety of the variety of commutative residuated lattices, we study congruences of NPc-lattices and some subvarieties of NPc-lattices.


INTRODUCTION
The well known disjunction property of intuitionistic propositional calculus asserts that a disjunction α ∨ β is provable if and only if α is provable or β is provable.The constructive character of disjunction 1 Dedicated to Newton A. C. Da Costa on his 80th birthday.Manuscrito -Rev. Int. Fil., Campinas, v. 34, n. 1, p. 99-114, jan.-jun. 2011. is not shared by negation: ¬(α ∧ β) in general does not imply ¬ α or ¬ β.This motivated D. Nelson (1949) to introduce constructive logic with strong negation (CLN) as an expansion of intuitionistic logic by a new negation symbol ∼.The propositional fragment of CLN can be axiomatized by adding to axioms of propositional positive intuitionistic logic the axioms: The deduction rules are modus ponens and substitution.
In the short paper Almukdad and Nelson (1984) it is observed that by deleting N 5 one obtains "a constructive logic which may be applied to inconsistent subject matter without necessarily generating a trivial theory."The system obtained by deleting N 5 is known as paraconsistent Nelson's logic.
Both, CLN and paraconsistent Nelson's logic are algebraizable.The corresponding algebraic structures are Nelson algebras and N4-lattices, respectively.Nelson algebras and N4-lattices can be represented as twist-structures of Heyting algebras and generalized Heyting algebras (also known as implicative lattices) respectively.This representation, due to Sendlewski (1990) for Nelson algebras and to Odinstov (2004) for N4-lattices, has been the main tool to study the algebraic semantics of CLN and its paraconsistent version.For details see the monograph Odintsov (2008).
Many important logics are particular cases of substructural logics (Galatos, Jipsen, Kowalski and Ono (2007)), i.e., logics that lack some of the three structural rules of contraction, weakening and exchange.
Due to this fact, in the recent years the study of substructural logics have been greatly developed to provide a common framework to treat and compare these different logics.Residuated lattices are the algebraic counterpart of substructural logics.With the aim of situating Nelson's logic within this framework, M. Spinks andR. Veroff (2008a, 2008b) showed that Nelson algebras are term equivalent to a class of residuated lattices.This class of residuated lattices is a variety, the variety of Nelson residuated lattices (see also Busaniche and Cignoli, 2010).Spinks and Veroff's result allows us to use the well-developed theory of residuated lattices to investigate the algebraic semantics of CLN.More important, it shows that CNL can be considered as an axiomatic extension of FL ew , the Full Lambek Calculus with Exchange and Weakening (see Galatos, Jipsen, Kowalski and Ono (2007)).
Following these ideas, in Busaniche and Cignoli (2009) we presented a quasivariety of commutative residuated lattices that serves as an algebraic semantics for paraconsistent Nelson's logic.The elements of this quasivariety are called NPc-lattices.As a matter of fact NPc-lattices form the algebraic semantics of a conservative expansion of paraconsistent Nelson's logic by a constant e that correspond to the unit of the underlying monoids of the residuated lattices.
In this note we improve the results of Busaniche and Cignoli (2009) by showing that NPc-lattices form a subvariety of the variety of commutative residuated lattices.This means that the mentioned expansion of paraconsistent Nelson's logic by the constant e is an axiomatic extension of FL e , the full Lambek calculus with exchange (Galatos, Jipsen, Kowalski and Ono (2007)).We show that the negative cone of an NPclattice is a generalized Heyting algebra, and that the congruence lattice of an NPc-lattice is isomorphic to the congruence lattice of its negative cone.This provides an interesting tool to investigate subvarieties of NPc-lattices.We prove that semisimple NPc-lattices form a subvariety of the variety of NPc-lattices, and that the representable NPc-lattices form a proper subvariety of the variety of semisimple NPc-lattices.Therefore semisimple NPc-lattices and representable NPc-lattices form the algebraic semantics of axiomatic extensions of FL e .It is worthwhile to notice that these algebraic results were obtained for the more general variety of Pc-lattices (see Definition 2.1), and hence do not depend on the representation as twist-structures of generalized Heyting algebras, in contrast with the results of Busaniche and Cignoli (2009).
All the notions from universal algebra used in this paper can be found in the book Burris and Sankappanavar (1981).

By a commutative residuated lattice we mean an algebra
) is a commutative monoid and the following residuation condition is satisfied: where x, y, z denote arbitrary elements of A and ≤ is the order given by the lattice structure.
The residuated condition (1) can be replaced by the following set of equations (see Hart, Rafter and Tsinakis (2002) Therefore commutative residuated lattices form a variety, that we shall denote by CRL.
If the underlying lattice of A ∈ CRL is distributive, we say that A is a commutative distributive residuated lattice.
It follows from R 1 that * is monotonic: 2) that A − is closed under the operations ∨, ∧, * , and if the binary operation ⇒ e is defined as then it is easy to check that A − = (A − , ∨, ∧, * , ⇒ e , e) is an integral commutative residuated lattice.
A residuated lattice with involution was defined in Busaniche and Cignoli (2009) as a commutative residuated lattice which satisfies the equation: and it was shown that if we define on a residuated lattice with involution A the unary operation ∼ by the prescription ∼ x = x ⇒ e for all x ∈ A, then the following properties are satisfied: Manuscrito -Rev.Int. Fil., Campinas, v. 34, n. 1, p. 99-114, jan.-jun. 2011.

NPc-LATTICES
Definition 2.1.A Paraconsistent residuated lattice (Pc-lattice for short), is a commutative distributive residuated lattice with involution A = (A, ∧, ∨, * , ⇒, e) satisfying the following equations, where ∼ x = x ⇒ e and x 2 = x * x: A Nelson Pc-lattice (NPc-lattice) is a Pc-lattice that satisfies the equation: We denote Pc and NPc the varieties of Pc-lattices and NPc-lattices, respectively.
Since ∼ e = e, by M 1 and M 3 , ( 8) is equivalent to: x ∧ e = y ∧ e and x ∨ e = y ∨ e imply x = y.(9) It was observed in Remark 4.9 in Busaniche and Cignoli (2009) that the lattice reduct of each NPc-lattice A is distributive.On the other hand it is well known, and easy to check, that for elements x, y, z of a distributive lattice, x ∨ y = x ∨ z and x ∧ y = x ∧ z imply y = z.Hence the requirement that the lattice be distributive turns the quasiequation (8) and the equation ( 7) redundant.Consequently the characterization of NPc-residuated lattices as a variety given here is equivalent to the definition of NPc-lattices given in Definition 4.1 in Busaniche and Cignoli (2009).Given a Pc-lattice A, its positive cone given by Because of M 1 , the set A = A − ∪ A + is symmetric with respect to e.
A generalized Heyting algebra (called implicative lattice by Odintsov (2003,2004)) is an integral residuated lattice that satisfies the equation: Notice that e is definable by x ⇒ x for any x ∈ H.For simplicity, when we refer to a generalized Heyting algebra we omit the operation * and we write simply H = (H, ∨, ∧, ⇒, e).Generalized Heyting algebras can be thought of as bottom-free reducts of Heyting algebras.
Since (5) implies that (10) holds in the negative cone of each Pclattice, we have: with the operations ∨, ∧, * , ⇒ given by (15) The reader can verify that the following equation holds in every A ∈ Pc (a proof is given in Lemma 4.2 of Busaniche and Cignoli ( 2009)).Hart, Rafter and Tsinakis (2002):

CONGRUENCES OF Pc-LATTICES
Theorem 3.1.The correspondence θ → S θ establishes an order isomorphism from the set Sub c (A) of convex subalgebras of A onto the set Cong(A) of congruences of A, when both sets are ordered by inclusion.
An implicative filter (i-filter for short) of an integral commutative residuated lattice A is a subset F ⊆ A such that e ∈ F and it is closed under modus ponens: x ∈ F and x ⇒ y ∈ F imply y ∈ F .Implicative filters can also be characterized as subsets of A that are nonempty, upwards closed and closed by * .It follows easily that implicative filters are precisely the convex subalgebras of integral commutative residuated lattices.Hence by Theorem 3.1, there is an order isomorphism from Cong(A) onto the set Filt(A) of i-filters of A, ordered by inclusion.
Theorem 3.2.Let A be a Pc-lattice.The correspondence φ : is an order isomorphism.2Proof.Let F be an i-filter of the integral residuated lattice Since F is upwards closed we get x ∈ F. We can conclude that φ is injective.
To check surjectivity, let S ∈ Sub c (A).First we see that F = S∩A − is an implicative filter of the negative cone of A. Clearly e ∈ S ∩ A − and if x, y ∈ S ∩ A − , then x * y ∈ S ∩ A − .To see that S ∩ A − is upwards closed, let x ∈ S ∩ A − and x ≤ y ≤ e.Then x * y ≤ e * e = e, and y ≤ x ⇒ e. Hence we have x ≤ y ≤ x ⇒ e and since S is convex we get y ∈ S ∩ A − .Now we prove that S = C(F ).The inclusion C(F ) ⊆ S follows immediately from the convexity of S. For the opposite inclusion, take s ∈ S. Since S is a subalgebra of A, the element h = s ∧ e ∧ (s ⇒ e) belongs to S ∩ A − .We have It is left as an easy exercise to corroborate that φ is order preserving. . . .Notice that the inverse of the isomorphism φ in the above theorem is the correspondence S → S ∩ A − .As an immediate corollary we get:

SEMISIMPLE AND REPRESENTABLE Pc-LATTICES
We first prove a result that will help us deal with semisimplicity.Proof.As an immediate consequence of Lemma 4.3, any semisimple algebra in Pc must be in V.
The algebra I(B) satisfies equation ( 17), because I(B) − is the reduct of a boolean algebra.Since this algebra generates V, every algebra in V satisfies equation ( 17).Due to Lemma 4.1, we can assert that the negative cone of every algebra in V is a boolean algebra.In particular, if C is a subdirectly irreducible algebra in V, then C − is a subdirectly irreducible boolean algebra.This means that C − is the two element boolean algebra.Thus C is a simple Pc-lattice.We can conclude that subdirectly irreducible algebras in V are simple.Hence all the elements of V are semisimple Pc-lattices.
The last statement of the theorem follows from the fact that I(B) is an NPc-lattice.
It is easy to see that P 3 generates a proper subvariety of the variety of semisimple Pc-lattices, characterized by ( 17) and the following Kleene equation: We are going to show that this variety coincides with the variety of representable Pc-lattices.
A residuated lattice is representable if it is a subdirect product of linearly ordered residuated lattices.Given a subvariety V ⊆ CRL, it is shown in §3 in Tsinakis and Wille (2006) that the representable residuated lattices in V form a subvariety of V characterized by equation ( 7) and the equation e ∧ ((x ⇒ y) ∨ (y ⇒ x)) = e. (20) We have already observed that (7) holds in any Pc-lattice.Therefore we will search for subvarieties of Pc that satisfy (20).
To achieve such an aim, we will investigate the possible structure of totally ordered Pc-lattices.Obviously the trivial Pc-lattice whose only element is e is totally ordered.
Theorem 4.5.The Pc-lattice P 3 is the only nontrivial totally ordered Pc-lattice.
Proof.Let L be a nontrivial totally ordered Pc-lattice.Obviously L = L − ∪ L + , thus the chain L must be symmetric with respect to e.
Then it can not be the case that L has only two elements.If L has three elements, the result of Lemma 4.2 yields that L ∼ = P 3 .
) and the quasiequation If x ∧ e = y ∧ e and ∼ x ∧ e = ∼ y ∧ e, then x = y.(8)

Corollary 3. 3 .
The lattices Cong(A) and Cong(A − ) are isomorphic.Theorem 3.2 provides a useful tool to analize some classes and subvarieties of Pc and NPc.Two of its most important consequences are summarized in the next lemma.Lemma 3.4.Let A be a Pc-lattice.Then:1.A issimple if and only if A − is the two-element Boolean algebra.2. A is subdirectly irreducible if and only if A − has a coatom.

Lemma 4. 1 .
Let A ∈ Pc.Then A − is the bottom-free reduct of a boolean algebra if and only if A satisfies the equation (((a ∧ e ⇒ b) ∧ e) ⇒ a) ∧ e = a ∧ e. (17) Proof.It is well known that the generalized Heyting algebra A − is the bottom-free reduct of a boolean algebra if and only if it satisfies the Peirce equation (a ⇒ e b) ⇒ e a = a.(18) Observe that for every pair of elements a, b ∈ A − , since equation (16) holds in A we get that (a ⇒ e b) ⇒ e a = ((a ∧ e) ⇒ e (b ∧ e)) ⇒ e (a ∧ e) = (((a ∧ e ⇒ b ∧ e) ∧ e) ⇒ a ∧ e) ∧ e = (((a ∧ e ⇒ b) ∧ e) ⇒ a ∧ e) ∧ e = Lemma 3.4 we know that C − is the two-element boolean algebra B.We will prove that C is a subalgebra of I(B).Since e is the greatest element in C − , without danger of confusion we can denote by ⊥ the other element in C − , thus C − = {⊥ < e}.After defining ∼ ⊥ = , we have that C − ∪ C + is the set{⊥ < e < }.Observe that this totally ordered set is order isomorphic to the lattice reduct of the algebraP 3 .Let c ∈ C be such that c / ∈ C − ∪ C + .Since c ∧ e isan element in C − less than e we must have c ∧ e = ⊥.Similarly c ∨ e = .If there were c, d / ∈ C − ∪ C + , then c ∧ e = d ∧ e and c ∨ e = d ∨ e.As previously mentioned, the distributivity of C implies the quasiequation (9), thus c = d.From the result of Lemma 4.2 we can conclude that either C ∼ = P 3 or C ∼ = I(B).Theorem 4.4.The class of semisimple Pc-lattices is the subvariety V of Pc generated by I(B), and it is characterized by equation (17).Moreover, the class of semisimple Pc-lattices coincides with the class of semisimple NPc-lattices.