## Servicios Personalizados

## Revista

## Articulo

## Indicadores

- Citado por SciELO
- Accesos

## Links relacionados

- Similares en SciELO

## Compartir

## Acta Nova

##
*versión On-line* ISSN 1683-0789

### RevActaNova. v.9 n.5-6 Cochabamba nov. 2020

**Artículo Científico**

**Special Sheffer-operators on a p-valued logic**

*Operadores de Sheffer especiales en una lógica p-valuada*

**Oscar R. Pino Ortiz**

Universidad Católica Boliviana, Cochabamba, Bolivia

Los Nogales 2030 Cbba,

Recibido: 12 de marzo 2020

Aceptado: 2 de septiembre 2020

**Abstract: **We studied the Sheffer-operators that can be written under the form 1 + *α*(*r,s*) where *α* is an associative, commutative and idempotent operator on *Z _{p}*[

*x, y*], with

*p*prime. We conjecture that such operators are always Sheffer operators. We show the conjecture is true for

*p*= 2 and

*p*= 3.

**Key words: **Logic, Multivalued, Sheffer.

**Resumen: **Se estudia los operadores de Sheffer que pueden escribirse bajo la forma 1 +*α*(*r,s*), donde *α* es una operación asociativa, conmutativa e idempotente sobre *Z _{p}*[

*x, y*], con

*p*primo. Conjeturamos que tales operadores son siempre operadores de Sheffer y mostramos que la conjetura es cierta para

*p*= 2 y

*p*= 3.

**Palabras Clave: **Lógica, Multivaluada, Sheffer.

**1** **Introduction**

In a previous article [1] we have shown that every operator in a p-valued logic can be written as a polynomial in *Z _{p} [x, *y]

*.*We have also remarked that three of the most known multivalued Sheffer-operators can be written as 1 +

*α*(

*x,*y), where

*α*is associative, commutative and idempotent on its variables. This is the case of Post-operator

*inc*(min(

*x*,

*y*)), the Webb-operator

*inc*(max(

*x*,

*y*)) and the Webb stroke function |, whose polynomial form is δ

^{1}.

For instance, if *p* = 3, the three mentioned operators are:

whose polynomial expressions^{2} are:

Of course, these three operators are not the only ones that have the properties indicated above. How many there are? Which are they? Are all the operators with these properties Sheffer-operators?

**2** **A theorem on special Sheffer-operators**

To answer these questions, we introduce an action of the *S _{p} *group on the set

*Z*[

_{p}*x, y*] defined by

Where *θ * *S _{p}. *It is clear that

*θ **

*γ*is in

*Z*[

_{p}*x, y*] if

*γ*is in

*Z*[

_{p}*x, y*].

^{3}

Let's take *S _{p} *=<

*ρ;*

*τ*

*>,*where

*ρ*is the first

*p*-cycle and

*τ*is the transposition between 0 and 1. For instance, if

*p*= 3 , we have

*ρ*(

*x*) = 1

**+**

*x*and

*τ*(

*x*) = 1 + 2x.

**Theorem 1 **Let be *θ * *S _{p} *and

*γ*Z

_{p}[

*x*,

*y*].

*θ * **γ* is a Sheffer operator if* γ** *it is.

**Proof**

For an operator* γ*(*x, y*) *Z _{p}*[

*x, y*], define

*Im*(

*γ*) in a recursive way:

1) x *Im*(*γ*)

2) *y ** Im*(*γ*)

3) *r,s * *Im*(*γ*) * γ*(*r,s*) * Im*(*γ*)

Then *γ* is a Sheffer-operator if and only if *Im*(*γ*) = *Z _{p}*[

*x*,

*y*]

*.*

Now, if* γ* is a Sheffer-operator^{4}, it is easy to see that *x*, *y* *Im*(*θ * **γ*). Indeed *θ * x = x *and *θ * y = y.*

Now let be *r* and *s* in *Im*(*γ*). Then* γ*(*r, s*) *Im*(*γ*). Since *γ* is a Sheffer operator we have that *θ*^{-1}(*r*) and *θ*^{-1}(*s*) are in *Im*(*y*). So we can use the identity:

That is if* γ*(*r, s*) *Im*(*γ*) then* γ*(θ^{-1}(*r*),θ^{-1}(*s*)) *Im*(*γ*) and then:

Like *θ*: *Z _{p}*[

*x, y*]

*Z*[

_{p}*x, y*] is a one to one map, whose inverse is θ

^{-1}

*,*we have

**Theorem 2**

If *α** ** Z _{p}*[

*x, y*] is associative, so

*θ**

*α*it is.

If* α * *Z _{p}*[

*x, y*] is commutative, so

*θ**

*α*it is.

If* α * *Z _{p}*[

*x, y*] is idempotent, so

*θ**

*α*it is.

**Proof**

1. Suppose that* α * *Z _{p}*[

*x, y*] is associative. It means

*α*(

*r,*

*α*(

*s,t*)) =

*α*(

*α*(

*r*,

*s*),

*t*) for all

*r, s, t*

*Z*[

_{p}*x*,

*y*].

Then

2. Suppose that *α* *Z _{p}*[

*x,y*] is commutative. It means

*α*(

*r,s*)

*=*

*α*(

*s,r*) for all

*r,s*

*Z*[

_{p}*x,y*].

3. Suppose that *α* *Z _{p}*[

*x,y*] is idempotent. It means

*α*(

*r,r*) =

*r*for all

*r*

*Z*[

_{p}*x*,

*y*].

**Corollary**

Let be

*A _{p} = *{

*α*

*Z*[

_{p}*x,y*]

*|*

*α*

*is asociative, comutative and idempotent*}.

*If **α** ** A _{p} *then

*θ **

*α*

*A*

_{p}.Cases **p**** = 3** and **p**** = 5**

For *p* = 3 we have nine polynomials of the form 1 +* α*(*x, *y) with *α* *A*_{3}. *α* must be one of:

We see that *α*_{1} = *α** _{|}*,

*α*

_{2}=*and*

*α*

_{7}=*α*

_{v}

*.*All nine 1 +

*α*

*(*

_{i}*x*,

*y*) are Sheffer operators. We have two orbits:

Using a computer, for *p* = 5 we have found 1065 Sheffer operators of this special form. We conjecture that if an operator has this special form then it is a Sheffer operator. This assumption is true for little values of *p* prime. For instance, for *p* = 2 we have only two Sheffer operators 1 + *xy* and 1 + *x* + *y* + *xy*, the Sheffer stroke and the Pierce arrow and, in this case, it is very easy to show that the conjecture is true. For *p* = 3 we are in a similar situation, because under the conditions imposed to *α* * A*_{3} we found that 1 +*α*(*x, y*) is one of the nine Sheffer operators listed above. Indeed...

**Theorem 3**

If *γ*(*x,y*) = 1** + ***α*(*x*,*y*) is an operator of a 3-valued logic with* α ** A*_{3},* γ* is a Sheffer operator.

**Proof**

Since* α *is commutative and idempotent, the matrix form^{5} of *α* is with *a**, b, c * *Z _{p}. *Of course, we have

*α*(0,1) =

*a*

*,*

*α*(0,2) =

*b*and

*α*(1,2) =

*c*.

There are 27 cases.

If* a **= 2 *then *b = 2 *and c = 2, because

By the same way we can show that:

if *b = *1 then *a* = 1 and c = 1

if *c = *0 then *a* = 0 and *b = *0

So we have the following three operators in *A*_{3}

On the other hand we can exclude all the operators failing the rules showed above. There are 1·3·3 + 2·1·3 + 2·2·1 - 3 = 16 of them. They all are not associative.

There are eight operators left to study. For them we have* a * {0; 1}*, b * {O;2}, *c* {1;2}.

We see there are two cases that are clearly not associative: *a** = *0, *b = 2, c = *1 and *a** = 1, b = *0, *c = 2 *because

The six operators remaining are:

They are all six associative.

We recognize the two orbits formed by the action of *S*_{3} on *A*_{3}*. *In the first orbit there is the well-known operator corresponding to *α** _{|}*(

*x,*y). Since 1 +

*α*

*(*

_{|}*x*,

*y*) is the Webb stroke and it is a Sheffer operator, all the operators in its orbit define also Sheffer operators.

In the second orbit we have another operator we know quite well:

It corresponds to (*x*, *y*). Like 1 + (*x*, *y*) is the Post-operator and it is a Sheffer operator we have again that all the operators in its orbit define Sheffer operators.

**Notes**

^{1} *δ *is a Sheffer-operator. Donald L. Webb showed it in 1935 [3]. We rewrote the proof under an algebraic point of view [2].

^{2} The way we build the polynomial expression of a logic operator is explained in detail in [2].

^{3} You can easily see that * thus defined is an action à* droite *of a group on a set.

^{4} Remember that a logic operator* γ* is called a Sheffer operator if and only if all the logic operators may be written using only *γ**.*

^{5} See [2].

**Bibliography**

[1] Pino O., Morales Z. (2015) *Un operador de Sheffer en la Lógica IGR _{3}. *Acta Nova, Vol 7, Nº1. Cochabamba, Bolivia.

[2] Pino O. (2018) *Un operador de Sheffer en la Lógica IGR _{p}. *Acta Nova, Vol 8, Nº4. Cochabamba, Bolivia.

[3] Webb D. L. (1935). *Generation of any n-valued logic by one binary operation*. Proceedings National Academy of Sciences. U.S.A. May 1935.

[4] Stojmenovic I., (1988). *On Sheffer symmetric functions in three valued logic*. Discrete Applied Mathematics 22, North-Holland. [ Links ]

[5] Foxley E., (1962) *The determination of all Sheffer functions in 3-valued logic, using a logical computer***. **Notre Dame Journal of Formal Logic, Volume III, Number 1. Nottingham, England.