Epimorphisms between finite 4−valued modal algebras A. V. Figallo, E. Pick and S. Saad Abstract In this paper we determine conditions for the existence of an epimorphism between two finite 4−valued modal algebras and state a method to obtain it. Furthermore, we obtain formulas which generalize those indicated by R. Sikorski for finite Boolean algebras ([1]), and by M. Abad and A. V. Figallo for finite 3−valued Lukasiewicz algebras ([2]).
AMS 2000 Subject Classification: Primary 08A35, 06D30. Secondary 03G25. Key words: Four valued modal algebras, tetravalent modal algebras, De Morgan algebra, epimorphisms, automorphisms.
1
Introduction and preliminaries
Tetravalent modal algebras were first considered by A. Monteiro and studied mainly by I. Loureiro, A.V. Figallo, A. Ziliani and P. Landini (see [3, 4, 5, 6, 7, 8, 9, 10, 11]). Later, in 2000, J.M. Font and M. Rius ([12]) showed their interest in the logics originated by the lattice aspects of these algebras. Following A. V. Figallo’s terminology, we call them four-valued modal algebras (or 4-valued modal algebras). In addition, the interesting 2013 work by M. Coniglio and M. Figallo (see [13]) has given, in our opinion, a new impulse to the profound study and development of the tetravalent modal algebras. For further information the reader interested in this subject is referred to the bibliography recommended below. Let us now recall that An algebra hA, ∧, ∨, ∼, 0, 1i of type (2, 2, 1, 0, 0) is a De Morgan algebra ([14]) if hA, ∧, ∨, 0, 1i is a bound distributive lattice with least element 0, greatest element 1, and ∼ satifies the equations ∼∼ x = x and ∼ (x ∨ y) = ∼ x ∧ ∼ y. It is known that a finite De Morgan algebra A is determined by its determinant system (Π(A), Ψ), where Π(A) is the ordered set of all prime elements of A and Ψ : Π(A) −→ Π(A) is an antitone involution of Π(A). Thus, Ψ has the following properties: (i) Ψ(Ψ(p)) = p, for each p ∈ Π(A), (ii) if p1 , p2 ∈ Π(A) and p1 ≤ p2 , then Ψ(p2 ) ≤ Ψ(p1 ). Moreover, the operation ∼ is given in the following way: W {p ∈ Π(A) : Ψ(p) 6≤ x}, if ∼x= 1, if
x 6= 0, x = 0.
In 1978 A. Monteiro defined the 4-valued modal algebras (or M4 -algebras) as algebras hA, ∧, ∨, ∼ , ∇, 0, 1i of type (2, 2, 1, 1, 0, 0), where hA, ∧, ∨, ∼, 0, 1i is a De Morgan algebra and the properties ∼ x ∨ ∇x = 1,
∼ x ∧ ∇x = x∧ ∼ x.
are verified. We assume the reader to be familiar with the theory of M4 -algebras as it is given in [3, 4]. In particular, in [4] is the demonstration of the following properties, which are used in this paper: x ≤ ∇x,
∇0 = 0,
∇∇x = ∇x,
∇(x ∨ y) = ∇x ∨ ∇y,
If the operator ∆ is defined by the formula ∆x =∼ ∇ ∼ x, then it is easy to see that ∆∇x = ∇x. In what follows, we only consider finite M4 -algebras. From [14] me have that the determinant system (Π(A), Ψ) of an M4 -algebra A has connected components of the three following types: x •
Type I:
......... ... ... ... ... ........ .
Type II:
Type III:
with Ψ(p) = p,
•.... p∗ ... ... .. ... ... . •..
with p < p∗ ,
p
with p and p∗ incomparable,
• .......................................... • ∗
p
Ψ(p) = p∗ and
Ψ(p) = p∗ .
p
The ordered set Π(A) has the following diagram: p∗t+1 •..... ........
x x • ··· • ··· p1
x • pt
.... ... ... ... ...
... .... ... .... ..... • ..
pt+1
p∗n
•..... ........
···
.... ... ... ... ...
.... ... ... .. ........ •
•..... ........
···
.... ... ... ... ...
... .... ... .... ..... • ..
pn
• ............................... •
pn+1 p∗n+1
···
• ............................... •
···
We will denote by B(A), T (A), and M (A) the ordered sets {p1 , · · · , pt }, {pt+1 , p∗t+1 , · · · , pn , p∗n }, {pn+1 , p∗n+1 , · · · , pk , p∗k }, respectively. Frequently we will write At,n−t,k−n instead of A. On the other hand, the operator ∇ is given by the formula W {p ∨ Ψ(p) : p ∈ Π(A), p ≤ x}, if ∇x = 0, if Then, 2
x 6= 0, x = 0.
• ................................. • p∗k
pk
pj , if p = pj and 1 ≤ j ≤ t, p∗j , if p = pj or p = p∗j and t + 1 ≤ j ≤ n, ∇p = pj ∨ Ψ(pj ), if p = pj or p = p∗j and n + 1 ≤ j ≤ k.
2
M4 -epimorphisms and M4 -functions
We will denote the sets Π(A) ∪ {∇p : p ∈ Π(A)} and M (A) ∪ {∇p : p ∈ M (A)} by R(A) and S(A), respectively. The lemma given below will be used in the proof of Lemma 2.5 Lemma 2.1 If p ∈ Π(A), then either p = ∇p or ∆p = 0 Proof. If p 6= ∇p, then p < ∇p, so p = pj with t + 1 ≤ j ≤ k. Then, W W q, ∼ pj = q= q∈Π(A)
Ψ(q)6≤pj
∇ ∼ pj = ∇
W q∈Π(A) q6=p∗ j
q∈Π(A) q6=p∗ j
q=
W q∈Π(A) q6=p∗ j
W
∇q =
(q ∨ Ψ(q)) =
W
q = 1.
q∈Π(A)
q∈Π(A) q6=p∗ j
Thus, ∼ ∇ ∼ pj = 0, that is ∆pj = ∆p = 0.
In what follows, we will denote with Epi(A, A0 ) the set of all M4 -epimorphisms from the M4 algebra A into the M4 -algebra A0 . Definition 2.1 A mapping f : R(A0 ) −→ R(A) is called an M4 -function if and only if the following conditions are satisfied: (M1) f is one to one, (M2) f (Ψ(p0 )) = Ψ(f (p0 ) for all p0 ∈ Π(A0 ) and f (p0 ) ∈ Π(A0 ), (M3) f (∇q 0 ) = ∇(f (q 0 )) for all q 0 ∈ R(A0 ). We will denote by F(A0 , A) the set of all M4 -functions from R(A0 ) into R(A). Lemma 2.2 If f ∈ F(A0 , A), A∗x = {q 0 ∈ R(A0 ) : f (q 0 ) ≤ x} and A0x = {p0 ∈ Π(A0 ) : f (p0 ) ≤ x}, then: (i) f is an isotone map, W W (ii) {q 0 : q 0 ∈ A∗x } = {p0 : p0 ∈ A0x }. Proof. (i) If q10 , q20 ∈ R(A0 ) and q10 < q20 , then ∇q10 = q20 . Thus, f (q10 ) ≤ ∇f (q10 ) = f (∇q10 ) = f (q20 ), and by (M1) we obtain f (q10 ) < f (q20 ). 3
W W W (ii) Let Nx = A∗x \A0x , a = {p0 : p0 ∈ A0x }, b = {q 0 : q 0 ∈ A∗x }, c = {r0 : r0 ∈ Nx }. It is easy to see that b = a ∨ c. We shall show that c ≤ a. Indeed, if r0 ∈ Nx , then there exists p0o ∈ Π(A0 ) such that r0 = p0o ∨ Ψ(p0o ). Hence, p0o ≤ r0 and ψ(p0o ) ≤ r0 . Since f (r0 ) ≤ x, we obtain f (p0o ) ≤ x and f (Ψ(p0o )) ≤ x, with p0o , Ψ(p0o ) ∈ Π(A), that is p0o , Ψ(p0o ) ∈ A0x . Hence, r0 ≤ a; thus, c ≤ a. Finally, b = a. Definition 2.2 Let f ∈ (A0 , A). F : A −→ A0 is an M -function associated with f if and only if F is defined by: W 0 0 {q : q ∈ A∗x }, if A∗x 6= ∅, F (x) = 0, if A∗x = ∅. We denote by M(A, A0 ) the set of all M -functions. The following lemma is used to prove that every M -function is an epimorphism. Lemma 2.3 (I) If p0 ∈ Π(A0 ) and f (Ψ(p0 )) 6≤ x, then Ψ(p0 ) 6≤ F (x). (II) If p ∈ Π(A), then F (p) = 0 or F (p) ∈ Π(A0 ). (III) If p ∈ Π(A), then F (∇p) = ∇F (p). Proof. W (I) Assume that Ψ(p0 ) ≤ F (x). Since F (x) = {s0 : s0 ∈ A0x } and Ψ(p0 ) ∈ Π(A0 ), it follows that there exists s0o ∈ A0x , such that Ψ(p0 ) ≤ s0o . From this we obtain f (Ψ(p0 ) ≤ f (s0o ); as a consequence, f (Ψ(p0 )) ≤ x, which contradicts the hypothesis. (II) Suppose F (p) 6= 0. There are two possibilities for p: (a) p is minimal in Π(A). As F (p) 6= 0, A0p 6= ∅, then there exists p0 ∈ Π(A0 ) such that f (p0 ) ≤ p. Since f (p0 ) ∈ Π(A) and p is minimal, then f (p0 ) = p must hold, and from (M1), A0p = {p0 } results. Thus, F (p) = p0 with f (p0 ) = p. (b) p is maximal and not minimal in Π(A). Then, p = ∇q with q ∈ Π(A) and q < p. From F (p) 6= 0 there exists p0 ∈ Π(A0 ) such that f (p0 ) ≤ p. If F (q) = 0 we get f (p0 ) = p, and then F (p) = p0 . If F (q) 6= 0, as in (a), there exists a unique q 0 ∈ Π(A0 ) such that f (q 0 ) = q. Moreover, (i) q 0 < ∇q 0 because if q 0 = ∇q 0 , then f (q 0 ) = f (∇q 0 ) = ∇f (q 0 ) = ∇q = p, which is not possible. Consequently, (ii) f (q 0 ) < f (∇q 0 ). From (i) it follows that q 0 is Type II or Type III; in the latter case Ψ(q 0 ) < ∇q 0 and therefore p = Ψ(q) = Ψ(f (q 0 )) = f (Ψ(q 0 )) < f (∇q 0 ) = ∇f (q 0 ) = ∇q = p, which is not possible. Then, q 0 is Type II, so that Ψ(q 0 ) = ∇q 0 , then ∇q 0 ∈ Π(A0 ), and thus (iii) f (∇q 0 ) ∈ Π(A0 ). From (ii) and (iii) it results f (∇q 0 ) = p. Therefore, A0p = {∇q 0 }, so F (p) = ∇q 0 . (III) Let p ∈ Π(A). (a) If F (p) 6= 0, then from (II) F (p) = p0 , with f (p0 ) = p. Thus, f (∇p0 ) = ∇f (p0 ) = ∇p; then, F (∇p) = ∇p0 = ∇F (p). 4
(b) If F (p) = 0, then ∇F (p) = ∇0 = 0. There are two cases: (b1 ) p = ∇p. Then, F (∇p) = F (p) = 0 = ∇F (p). (b2 ) p < ∇p and F (∇p) 6= 0, then: (b21 ) if ∇p ∈ Π(A), then F (∇p) = p0 with f (p0 ) = ∇p, and f (∇p0 ) = ∇f (p0 ) = ∇∇p = ∇p = f (p0 ); therefore, by (M1) ∇p0 = p0 . Moreover, it is clear that p0 is a minimal prime, because if q 0 < p0 , then f (q 0 ) < f (p0 ) = ∇p, which is not possible. Then, ∇p0 = p0 is minimal, and as a consequence, p0 = Ψ(p0 ). Thus, ∇p = f (p0 ) = f (Ψ(p0 )) = Ψ(f (p0 )) = Ψ(∇p), which is a contradiction. Then, F (∇p) = 0 holds; therefore, F (∇p) = ∇F (p) (b22 ) if ∇p ∈ / Π(A), then p = f (p0 ) is Type III; therefore, F (p) = p0 and F (∇p) = p0 ∨ Ψ(p0 ) = ∇p0 = ∇F (p). Lemma 2.4 If F(A0 , A) 6= ∅, then M(A, A0 ) ⊆ Epi(A, A0 ). Proof. Let f ∈ F(A0 , A) and F ∈ M(A, A0 ) be the M -function associated by f . (a) It is obvious that F (1) = 1 and F (0) = 0. (b) F (x ∨ y) = F (x) ∨ F (y). Actually, it is easy to see that A0x ∪ A0y ⊆ A0x∨y ; moreover, p0 ∈ A0x∨y if and only if f (p0 ) ≤ x ∨ y. Since f (p0 ) ∈ Π(A0 ), f (p0 ) ≤ x or f (p0 ) ≤ y must hold; hence, f (p0 ) ∈ A0x or f (p0 ) ∈ A0y , that is p0 ∈ A0x ∪ A0y ; then A0x∨y = A0x ∪ A0y . Finally, W W W F (x ∨ y) = A0x∨y = (A0x ∪ A0y ) = {r0: f (r0 ) ≤ x or f (r0 ) ≤ y} = F (x) ∨ F (y). (c) F (∼ x) =∼ F (x) for all x ∈ A. Indeed, (c1 ) F (∼ x) ≤∼ F (x). It is sufficient to show that, if q 0 ∈ A0∼x , then q 0 ≤∼ F (x). If qo0 ∈ A0∼x , then qo0 ∈ Π(A0 ) and f (qo0 ) ≤ ∼ x = ∨{p ∈ π(A) : Ψ(p) 6≤ x}. Hence, there exists po ∈ Π(A) such that f (qo0 ) ≤ po , and Ψ(po ) 6≤ xo . Then, Ψ(po ) ≤ Ψ(f (qo0 )) holds. Taking into account that f (Ψ(qo0 )) = Ψ(f (qo0 )), by Lemma 2.3(I), we have that Ψ(qo0 ) 6≤ F (x); then qo0 ≤∼ F (x). (c2 ) ∼ F (x) ≤ F (∼ x). Let p0 ∈ Π(A). The proof is a consequence of the fact that the following conditions are pairwise equivalent: (1) p0 ≤∼ F (x). (2) Ψ(p0 ) 6≤ F (x). (3) f (Ψ(p0 )) 6≤ x. (4) Ψ(f (p0 )) 6≤ x. (5) f (p0 ) ≤∼ x. (6) p0 ≤ F (∼ x). (d) F (∇x) = ∇F (x), for all x ∈ A, which is inmediate consequence of (III), ∇(x ∨ y) = ∇x ∨ ∇y and (b). (e) F is clearly onto. Lemmas 2.5 and 2.6 are necessary to show that all M4 -epimorphisms are an M4 -function M. Lemma 2.5 If h ∈ Epi(A, A0 ) and q 0 ∈ R(A0 ), then there exists a unique q ∈ R(A) such that h(q) = q 0 . 5
Proof. (1) (a) If q 0 ∈ Π(A0 ), then there is q ∈ Π(A) such that h(q) = q 0 . Indeed, let q 0 ∈ Π(A0 ); since h is onto, there exists x ∈ A such that h(x) = q 0 ; therefore h−1 ({q 0 }) 6= ∅, and then, l V h−1 ({q 0 }) = {x1 , · · · , xl }. If q = xi , then, h(q) = q 0 and q 6= 0. If q = y ∨ z, then i=1
y ≤ q, z ≤ q and q 0 = h(q) = h(y) ∨ h(z); since q 0 ∈ Π(A0 ), q 0 = h(y) or q 0 = h(z) must hold. If q 0 = h(y), then y ∈ h−1 ({q 0 }); therefore y = xj for some j ∈ {1, · · · , l}; hence, q ≤ y and so q = y. Then q = y or q = z. Thus, q ∈ Π(A). (b) If q ∈ Π(A) and h(q) = q 0 , then q is unique. Indeed, if p ∈ h−1 ({q 0 }) ∩ Π(A), then p ∈ Π(A) and h(p) = q 0 . Accordlingly, q is the smallest r ∈ Π(A) such that h(r) = q 0 , q ≤ p. If q < p, then p = ∇q; by Lemma 2.1, ∆q = 0 must hold. Furthermore, q 0 = h(p) = h(∇p) = ∇h(p) = ∇q 0 , and then ∆q 0 = ∆h(p) = ∆∇q 0 = ∇q 0 . On the other hand, ∆q 0 = ∆h(q) = h(∆q) = 0, which is a contradiction. (2) If q 0 ∈ R(A0 )\Π(A0 ), then there is p0 ∈ Π(A0 ) such that q 0 = p0 ∨ Ψ(p0 ); by 1. there is a unique p ∈ Π(A) such that h(p) = p0 . Let q = p ∨ Ψ(p); it is easy to prove that p and p0 are of the same type. So q 0 = ∇p0 = ∇h(p) = h(∇p) = h(p ∨ Ψ(p)) = h(p) ∨ h(Ψ(p)). In addition, the following holds: q 0 = ∇p0 = p0 ∨ Ψ(p0 ) = h(p ∨ Ψ(h(p)), giving the result that follows: h(p) ∨ h(Ψ(p)) = h(p) ∨ Ψ(h(p))
(i)
Moreover, h(p) ∧ Ψ(h(p)) = p0 ∧ Ψ(p0 ) = 0; h(p) ∧ h(Ψ(p)) = h(p ∧ Ψ(p)) = 0; then h(p) ∧ h(Ψ(p)) = h(p) ∧ Ψ(h(p))
(ii)
From (i) and (ii), it follows that h(Ψ(p)) = Ψ(h(p)). It is evident that q ∈ R(A) is unique and h(q) = q 0 . Definition 2.3 Let h ∈ Epi(A, A0 ). The function f : R(A0 ) −→ R(A) is induced by h if f is defined by f (q 0 ) = q, if and only if h(q) = q 0 . Lemma 2.6 Every function induced by an M4 -epimorphism is an M4 -function. Proof. f be the function induced by the M4 -epimorphism h. (M1) It is easy to prove that f is one to one. (M3) Let f (p0 ) = p, with p0 = h(p). Therefore, h(∇p) = ∇h(p) = ∇p0 ; consequently, f (∇p0 ) = ∇p, i.e. f (∇p0 ) = ∇f (p0 ). As a consequence of (M1) and (M3), f is an order-preserving function. (M2) Let p0 ∈ Π(A0 ), f (p0 ) = q with h(q) = p0 . (1) If p0 is Type I, then p0 = Ψ(p0 ) = ∇p0 , and (a) if q < Ψ(q), since h(q) ∈ Π(A) and h is one to one on the set of all prime p that h(p) 6= 0, h(q) < h(Ψ(q)) must hold; then p0 = h(q) < h(Ψ(q)) = h(∇q) = ∇h(q) = ∇p0 , which is a contradiction. 6
(b) if Ψ(q) < q, then h(Ψ(q)) ≤ h(q) = p0 . If h(Ψ(q)) = p0 , then f (p0 ) = Ψ(q); since f es one to one must be q = Ψ(q), which is a contradiction. So h(Ψ(q)) < p0 and consequently h(Ψ(q)) = 0 and then p0 = ∇p0 = ∇h(q) = h(∇q) = h(∇Ψ(q)) = ∇h(Ψ(q)) = ∇0 = 0, which is not possible. This shows that in this case Ψ(q) 6< q and q 6< Ψ(q); as a result Ψ(f (p0 )) = f (p0 ), so that either Ψ(f (p0 )) = f (Ψ(p0 )), or Ψ(q) and q are not comparable, which is not possible since q = f (p0 ) = f (∇p0 ) = ∇f (p0 ) = ∇q. (2) If p0 is Type II, then either p0 < Ψ(p0 ) or Ψ(p0 ) < p0 . In the first case f (p0 ) < f (Ψ(p0 )) and therefore Ψ(f (p0 )) = f (Ψ(p0 )). A similar argument is valid if Ψ(p0 ) < p0 . (3) If p0 is Type III, then f (p0 ) is Type III, and it is not comparable with f (Ψ(p0 )); consequently, f (Ψ(p0 )) = Ψ(f (p0 )). Lemma 2.7 Epi(A, A0 ) ⊆ M(A, A0 ). Proof. If Epi(A, A0 ) = ∅, the proof is obvious. Let h ∈ Epi(A, A0 ), f the function induced by h, and F the = 0 = F (0). Let x ∈ A, x 6= 0. W M -function associated to f . It is clear that h(0) W Then, x = {p : p ∈ Π(A), p ≤ x} and consequently, h(x) = {h(p) : p ∈ Π(A), p ≤ x}. W W For each p ∈ Π(A), p ≤ x, let p0 = h(p) ∈ Π(A0 ). Then h(x) = {p0 : p0 ∈ Π(A0 ), f (p0 ) ≤ x} = {p0 : p0 ∈ A0x } = F (x). Thus, h = F ; as a consequence, h ∈ M(A, A0 ). Teorema 2.1 Epi(A, A0 ) = M(A, A0 ). Proof. It follows from Lemmas 2.4 and 2.7.
Teorema 2.2 | Epi(A, A0 ) |=| F(A0 , A) |. Remarks 2.1 Let A = At,m−t,k−m , A0 = As,n−s,r−n . Then, (i) F (A0 , A) 6= ∅ if only if s ≤ t, n − s ≤ m − t, and r − n ≤ k − m. (ii) The following holds | B1 (A0 , A) |= Vts ,
n−s | T1 (A0 , A) |= Vm−t ,
| S1 (A0 , A) |=
r−n−1 Y
(2 · (k − m) − 2 · i),
i=0
where Vpq =
p! (p − q)!
if p > q,
0,
otherwise, 0
0
B1 (A , A) is the set of all injective functions f ∈ B(A)B(A ) , T1 (A0 , A) is the set of all injective 0 functions f ∈ T (A)T (A ) such that f (Ψ(p0 )) = Ψ(f (p0 )) for each p ∈ T (A), S1 (A0 , A) is the 0 set of all injective functions f ∈ S(A)S(A ) such that f (Ψ(p0 )) = Ψ(f (p0 )) for all p0 ∈ M (A0 ), 0 0 0 0 and f (∇(q )) = ∇f (q ) for all q ∈ S(A ). Teorema 2.3 If 1 ≤ s ≤ t, 1 ≤ n − s ≤ m − t, and 1 ≤ r − n ≤ k − m, then |
Epi(At,m−t,k−m , A0s,n−s,r−n )
r−n
|= 2
·
Vts
·
n−s Vm−t
·
r−n−1 Y i=0
7
(k − m − i)
Proof. If f ∈ F(A0 , A), f1 = f|B(A0 ) , f2 = f|T (A0 ) and f3 = f|S(A0 ) , then the map f −→ (f1 , f2 , f3 ) is a bijective correspondence between F (A0 , A) and B1 (A0 , A) × T1 (A0 , A) × S1 (A0 , A). From this, Theorem 2.1 and Remarks 2.1 this theorem follows. An immediate consequence of theorem 2.3 is Teorema 2.4 | Aut(At,m−t,k−m ) |= 2k−m · t! · (m − t)! · (k − m)!
References [1] R. Sikorski. On the Inducing of Homomorphisms by Mappings, Fundamenta Mathematicae 36 (1949), 7–22. [2] M. Abad, A. V. Figallo. On the Lukasiewicz homomorphisms. Instituto de Ciencias B´ asicas, 1992, UNSJ, 1-12. [3] I. Loureiro. Axiomatisation et proprietes des alg`ebres modales tetravalentes. C.R. Acad. Sc. Paris, t. 295 (22 novembre 1982), Serie I, 555–557 [4] I. Loureiro. Finite Tetravalent Modal Algebras, Revista de la Uni´on Matem´atica Argentina, 31, Nro 4 (1984), 187–191. [5] I. Loureiro. Prime spectrum of a tetravalent modal algebras, Notre Dame Journal of Formal Logic 24 (1983), 389–394. [6] I. Loureiro. Finite Tetravalent Modal Algebras, Revista de la Uni´on Matem´atica Argentina 31 (1984), 187–191. [7] A. V. Figallo. Notes on generalized N −lattices, Revista de la Uni´on Matem´atica Argentina 35 (1990), 61–65. [8] A. V. Figallo. On the congruence in four–valued modal algebras, Portugaliae Mathematica 49 (1992), 249–261. [9] A. V. Figallo and A. Ziliani. Symmetric tetra–valued modal algebras, Notas de la Sociedad Matem´ atica de Chile 10, 1 (1991), 133–141. [10] A. Ziliani. On axioms and properties of monadic four–valued algebras, Actas del Tercer Congreso Dr. A.R.R. Monteiro, Instituto de Matem´atica, U. N. del Sur, Argentina (1995), 69–78. [11] A. V. Figallo y P. Landini. On generalized I−algebras and modal 4−valued algebras, Reports on Mathematical Logic 29 (1995), 3–18. [12] J.M. Font and M. Rius. An abstract algebraic logic approach to tetravalent modal logics. Journal of Symbolic Logic, v. 65, n. 2 (2000), 481–518. [13] M.E. Coniglio and M. Figallo. Hilbert-style Presentations of Two Logics Associated to Tetravalent Modal Algebras. Studia Logica. First published online: July 16, 2013. DOI: 10.1007/s11225-013-9489-0. [14] A. Monteiro. Matrices de Morgan caracteristiques pour le calcul propositionnel classique. An. Acad. Brasil. Ciˆen. 52 (1960), 1–7. 8
Aldo V. Figallo Instituto de Ciencias B´ asicas, Universidad Nacional de San Juan. 5400 San Juan, Argentina. e-mail:
[email protected] Elda Pick Facultad de Ingeniera, Universidad Nacional de San Juan. 5400 San Juan, Argentina. e-mail:
[email protected] Susana Saad Instituto de Ciencias B´ asicas, Facultad de Filosofa, Universidad Nacional de San Juan. 5400 San Juan, Argentina. e-mail:
[email protected]
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