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,

opino@ucb.edu.bo

Recibido: 12 de marzo 2020
Aceptado: 2 de septiembre 2020

 

 

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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.

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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.

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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 δ¹.

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

whose polynomial expressions² 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-operator⁴, 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 A₃. α must be one of:

We see that α₁ = α_|, α₂ = and α₇ = 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 A₃ 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 A₃, γ is a Sheffer operator.

Proof

Since α is commutative and idempotent, the matrix form⁵ 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 A₃

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 S₃ on A₃. 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 IGR₃. 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.