1. Introduction
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] -[10] ). Later, in 2000, J. M. Font and M. Rius [11] 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 [12] ) 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
of type
is a De Morgan algebra [13] if
is a bounded distributive lattice with least element 0, greatest element 1, and
satifies the equations
and
.
It is known that a finite De Morgan algebra
is determined by its determinant system
, where
is the ordered set of all prime elements of
and
is an antitone involution of
. Thus,
has the following properties:
(i)
, for each
(ii) if
and
, then
.
Moreover, the operation
is given in the following way:
![](https://www.scirp.org/html/htmlimages\9-5300631x\62237029-db6e-4c77-b98b-96e2540f2a86.png)
In 1978 A. Monteiro defined the 4-valued modal algebras (or
-algebras) as algebras
of type
, where
is a De Morgan algebra and the properties
![](https://www.scirp.org/html/htmlimages\9-5300631x\2e14d8df-aa89-498d-a413-cad2eda33b12.png)
are verified.
We assume the reader to be familiar with the theory of
-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:
![](https://www.scirp.org/html/htmlimages\9-5300631x\aa9cc2e2-8dc4-4380-a936-96e7d750c271.png)
If the operator
is defined by the formula
, then it is easy to see that
.
In what follows, we only consider finite
-algebras.
From [13] me have that the determinant system
of an
-algebra
has connected components of the three following types:
![](https://www.scirp.org/html/htmlimages\9-5300631x\b0e0628f-5d69-4432-9dfc-2388d75a7e57.png)
![](https://www.scirp.org/html/htmlimages\9-5300631x\8dae245b-88da-408c-b0ad-1db3563b5981.png)
![](https://www.scirp.org/html/htmlimages\9-5300631x\b61c948f-cbc9-4e42-aeac-4d30f66efef6.png)
The ordered set
has the following diagram:
![](https://www.scirp.org/html/htmlimages\9-5300631x\335c12fb-114e-485b-9438-6c3b52a0cef6.png)
We will denote by
,
, and
the ordered sets
,
,
, respectively.
Frequently we will write
instead of
.
On the other hand, the operator
is given by the formula
![](https://www.scirp.org/html/htmlimages\9-5300631x\1bf0b8c6-1e31-4eb3-93ee-f3296a7febe4.png)
Then,
![](https://www.scirp.org/html/htmlimages\9-5300631x\4394a5d8-c652-4d61-b65c-a742f413e597.png)
2. M4-Epimorphisms and M4-Functions
We will denote the sets
and
by
and
, respectively.
The lemma given below will be used in the proof of Lemma 2.5 Lemma 2.1 If
, then either
or ![](https://www.scirp.org/html/htmlimages\9-5300631x\f29db448-594e-4c3f-af86-96caa7a1e82a.png)
Proof. If
, then
, so
with
. Then,
,
.
Thus,
, that is
. In what follows, we will denote with
the set of all
-epimorphisms from the
-algebra
into the
-algebra
.
Definition 2.1 A mapping
is called an
-function if and only if the following conditions are satisfied:
(M1)
is one to one(M2)
for all
and
(M3)
for all
.
We will denote by
the set of all
-functions from
into
.
Lemma 2.2 If
,
and
, then:
(i)
is an isotone map,
(ii)
.
Proof.
(i) If
and
, then
. Thus,
, and by (M1) we obtain
.
(ii) Let
,
,
,
. It is easy to see that
. We shall show that
. Indeed, if
, then there exists
such that
. Hence,
and
. Since
, we obtain
and
, with
, that is
. Hence,
; thus,
. Finally,
.
Definition 2.2 Let
.
is an
-function associated with
if and only if
is defined by:
![](https://www.scirp.org/html/htmlimages\9-5300631x\97df7562-a178-49fa-ac89-acaed510c61e.png)
We denote by
the set of all
-functions.
The following lemma is used to prove that every
-function is an epimorphism.
Lemma 2.3
(I) If
and
, then
.
(II) If
, then
or
.
(III)If
, then
.
Proof.
(I) Assume that
. Since
and
, it follows that there exists
, such that
. From this we obtain
; as a consequence,
, which contradicts the hypothesis.
(II) Suppose
. There are two possibilities for
:
(a)
is minimal in
. As
,
, then there exists
such that
. Since
and
is minimal, then
must hold, and from (M1),
results. Thus,
with
.
(b)
is maximal and not minimal in
. Then,
with
and
. From
there exists
such that
. If
we get
, and then
. If
, as in (a), there exists a unique
such that
. Moreover, (i)
because if
, then
, which is not possible. Consequently, (ii)
.
From (i) it follows that
is Type II or Type III; in the latter case
and therefore
, which is not possible. Then,
is Type II, so that
, then
, and thus (iii)
. From (ii) and (iii) it results
. Therefore,
, so
.
(III) Let
.
(a) If
, then from (II)
, with
. Thus,
; then,
.
(b) If
, then
. There are two cases:
(b1)
. Then,
.
(b2)
and
, then:
(b21) if
, then
with
, and
; therefore, by (M1)
. Moreover, it is clear that
is a minimal prime, because if
, then
, which is not possible. Then,
is minimal, and as a consequence,
. Thus,
, which is a contradiction. Then,
holds; therefore, ![](https://www.scirp.org/html/htmlimages\9-5300631x\a47e38fc-d3e2-4b49-9168-5b537731adb8.png)
(b22) if
, then
is Type III; therefore,
and
.
Lemma 2.4 If
, then
.
Proof. Let
and
be the
-function associated by
.
(a) It is obvious that
and
.
(b)
. Actually, it is easy to see that
; moreover,
if and only if
. Since
,
or
must hold; hence,
or
, that is
; then
. Finally,
.
(c)
for all
. Indeed,
(c1)
.
It is sufficient to show that, if
, then
.
If
, then
and
. Hence, there exists
such that
, and
. Then,
holds. Taking into account that
, by Lemma 2.3(I), we have that
; then
.
(c2)
.
Let
. The proof is a consequence of the fact that the following conditions are pairwise equivalent:
(1)
(2)
(3) ![](https://www.scirp.org/html/htmlimages\9-5300631x\d1237127-4144-4d8b-975d-f813de9bfa57.png)
(4)
(5)
(6) ![](https://www.scirp.org/html/htmlimages\9-5300631x\38f156db-c48f-4964-8d9e-e3e6ca652088.png)
(d)
, for all
, which is inmediate consequence of (III),
and (b).
(e)
is clearly onto.
Lemmas 2.5 and 2.6 are necessary to show that all
-epimorphisms are an
-function M.
Lemma 2.5 If
and
, then there exists a unique
such that
.
Proof.
(1) (a) If
, then there is
such that
Indeed, let
; since
is onto, there exists
such that
; therefore
, and then,
. If
, then,
and
. If
, then
,
and
; since
,
or
must hold. If
, then
; therefore
for some
; hence,
and so
. Then
or
. Thus,
.
(b) If
and
, then
is unique. Indeed, if
, then
and
. Accordlingly,
is the smallest
such that
,
. If
, then
; by Lemma 2.1,
must hold. Furthermore,
, and then
. On the other hand,
, which is a contradiction.
(2) If
, then there is
such that
; by 1. there is a unique
such that
. Let
; it is easy to prove that
and
are of the same type. So
.
In addition, the following holds:
, giving the result that follows:
(i)
Moreover,
;
; then
(ii)
From (i) and (ii), it follows that
. It is evident that
is unique and
.
Definition 2.3 Let
. The function
is induced by
if
is defined by
, if and only if
.
Lemma 2.6 Every function induced by an
-epimorphism is an
-function.
Proof.
be the function induced by the
-epimorphism
.
(M1) It is easy to prove that
is one to one.
(M3) Let
, with
. Therefore,
; consequently,
, i.e.
.
As a consequence of (M1) and (M3),
is an order-preserving function.
(M2) Let
,
with
.
(1) If
is Type I, then
, and
(a) if
, since
and
is one to one on the set of all prime
that
,
must hold; then
, which is a contradiction.
(b) if
, then
. If
, then
; since f es one to one must be
, which is a contradiction. So
and consequently
and then
, which is not possible.
This shows that in this case
and
; as a result
, so that either
, or
and
are not comparable, which is not possible since
.
(2) If
is Type II, then either
or
. In the first case
and therefore
. A similar argument is valid if
.
(3) If
is Type III, then
is Type III, and it is not comparable with
; consequently,
.
Lemma 2.7
.
Proof. If
, the proof is obvious. Let
,
the function induced by
, and
the
-function associated to
. It is clear that
. Let
,
. Then,
and consequently,
. For each
,
, let
. Then
. Thus,
; as a consequence,
. Teorema 2.1
.
Proof. It follows from Lemmas 2.4 and 2.7. Teorema 2.2
.
Remarks 2.1 Let
,
. Then(i)
if only if
,
, and
.
(ii) The following holds
![](https://www.scirp.org/html/htmlimages\9-5300631x\5b47d6e8-a20c-4adc-91fc-1c2ea5823ecd.png)
where ![](https://www.scirp.org/html/htmlimages\9-5300631x\a8cea3c1-e94f-4bc1-bed1-373a3f72014c.png)
is the set of all injective functions
,
is the set of all injective functions
such that
for each
,
is the set of all injective functions
such that
for all
, and
for all
.
Teorema 2.3 If
,
, and
, then
![](https://www.scirp.org/html/htmlimages\9-5300631x\437f9936-4902-47df-b5d8-97dbe6de9b90.png)
Proof. If
,
,
and
, then the map
is a bijective correspondence between
and
. From this, Theorem 2.1 and Remarks 2.1 this theorem follows.
An immediate consequence of theorem 2.3 is
Teorema 2.4 ![](https://www.scirp.org/html/htmlimages\9-5300631x\27842367-f729-4c0e-885a-ee4a5e6f8053.png)
,
AMS 2000 Subject Classification
Primary 08A35, 06D30. Secondary 03G25.