Startseite The Sheffer stroke operation reducts of basic algebras
Artikel Open Access

The Sheffer stroke operation reducts of basic algebras

  • Tahsin Oner EMAIL logo und Ibrahim Senturk
Veröffentlicht/Copyright: 13. Juli 2017
Artikel downloaden (PDF)
Open Mathematics
Aus der Zeitschrift Open Mathematics Band 15 Heft 1

Abstract

In this study, a term operation Sheffer stroke is presented in a given basic algebra 𝒜 and the properties of the Sheffer stroke reduct of 𝒜 are examined. In addition, we qualify such Sheffer stroke basic algebras. Finally, we construct a bridge between Sheffer stroke basic algebras and Boolean algebras.

Keywords: Basic algebras; Sheffer stroke reduct; Lattice with antitone involutions
MSC 2010: 06C15; 03G25; 06D30

1 Introduction

Take into consideration the problem of expressing equational concepts as simply as possible with the least number of operations or the least number of axioms and so forth. For an example about the least number of axioms, Tarski solved the related problem for Abelian groups with the single axiom x/(y/(z/(x/y))) = z in terms of the division operation in 1938 [1]. As a typical example, we tackle the use of Sheffer stroke operation in algebraic structures. The Sheffer stroke term operation was firstly given by H. M. Sheffer in 1913 [2]. He proved that all Boolean functions could be transplanted to a single binary operation for term operations. In recent years, the problem for Boolean algebra was solved with a single axiom in terms of the Sheffer Stroke operation [3].

The reduction attempts interest mathematicians who want to use less operations or axioms or formulas for the structures under their considerations. The first implication reduct of Boolean algebras connectives was studied by J. C. Abbott [4] under the name implication algebra. Once the logic of quantum mechanics was axiomatized by means of orthomodular lattices, Abbott obtained implication reducts of orthomodular lattices, called orthoimplication algebras in [5]. Later on, this work was generalized for implication reducts of ortholattices by Chajda and Halaš [6], and by Chajda [7] to orthomodular lattices but without the compatibility condition in Chajda, Halaš and Länger [8].

Basic algebras were introduced in Chajda and Emanovský [9], see also Chajda [10] and Chajda et al. [11] and [12, 13] for further information. Basic algebras are an important concept used in different non-classical logics since they contain orthomodular lattices ℒ = (L; ∨, ∧,⊥, 0, 1), where x ⊕ y = (x ∧ y⊥)∨ y and ¬x = x⊥, and constitute as well as provide an axiomatization of the logic of quantum mechanics along with MV-algebras [14], which get an axiomatization of many-valued Łukasiewicz logics; see Chajda [15] and Chajda et al. [16].

Given that the connective Sheffer stroke operation plays a central role in all mentioned logics above, in general, we would like to characterize this operation in basic algebras.

2 Preliminaries

The following fundamental notions are taken from [17] and [18].

Definition 2.1

A bounded lattice is an algebraic structure ℒ = (L; ∨, ∧, 0, 1) such that ℒ = (L; ∨, ∧) is a lattice having the following properties:

  1. for all x ∈ L, x ∨ 1 = 1 and x ∧ 1 = x,

  2. for all x ∈ L, x ∨ 1 = x and x ∧ 0 = 0.

The elements 0 and 1 are called the least element and the greatest element of the lattice, respectively.

Definition 2.2

Let ℒ = (L; ∨, ∧) be a lattice. A mapping x ↦ x⊥is called an antitone involution if it satisfies the following:

  1. x ≤ y implies y⊥ ≤ x⊥ (antitone),

  2. x⊥⊥ = x (involution).

Definition 2.3

Let ℒ be a bounded lattice with an antiotone involution. If it satisfies

x∨x⊥=1andx∧x⊥=0,

then x⊥ is the complement of x and ℒ = (L; ∨, ∧, ⊥, 0, 1) is an ortholattice.

Lemma 2.4

Let ℒ = (L; ∨, ∧, ⊥) be a lattice with antitone involution. Then it satisfies the following De Morgan laws:

x⊥∨y⊥=(x∧y)⊥andx⊥∧y⊥=(x∨y)⊥.

Definition 2.5

([19]). Let 𝒜 = (A, |) be a groupoid. The operation | is said to be a Sheffer stroke operation if it satisfies the following conditions:

(S1) x|y = y|x,

(S2) (x|x)|(x|y) = x,

(S3) x|((y|z)|(y|z)) = ((x|y)|(x|y))|z,

(S4) (x|((x|x)|(y|y)))|(x|((x|x)|(y|y))) = x.

If additionally it satisfies the identity

(S5) y|(x|(x|x)) = y|y, it is called an ortho Sheffer stroke operation.

Lemma 2.6

([19]). Let 𝒜 = (A, |) be a groupoid. The binary relation ≤ defined on A as below

x≤yifandonlyifx|y=x|x

is an order on A.

Lemma 2.7

([19]). Let | be a Sheffer stroke operation on A and ≤ the induced order of 𝒜 = (A, |). Then

  1. x ≤ y if and only if y|y ≤ x|x,

  2. x|(y|(x|x)) = x|x is the identity of 𝒜,

  3. x ≤ y implies y|z ≤ x|z for all z ∈ A,

  4. a ≤ x and a ≤ y imply x|y ≤ a|a.

In order to obtain a construction of Sheffer stroke reduction of basic algebras, we firstly give the definition of a basic algebra:

Definition 2.8

([20]). A basic algebra is an algebra 𝒜 = (A; ⊕, ¬, 0) of type (2, 1, 0) which satisfies the following axioms:

(BA1) x ⊕ 0 = x,

(BA2) ¬¬x = x,

(BA3) ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x,

(BA4) ¬(¬(¬(¬x ⊕ y) ⊕ y) ⊕ z) ⊕ (x ⊕ z) = ¬0.

As shown in [20], every basic algebra can be thought alternatively as a bounded lattice with section antitone involutions. Now, we are dealing with in the case when for an element x of a basic algebra 𝒜 the negation ¬x is a complement of x in the induced lattice ℒ(𝒜). Then we can give the following lemma and its corollary:

Lemma 2.9

([20]). Let 𝒜 = (A; ⊕, ¬, 0) be a basic algebra and x ∈ A. Then ¬x is a complement of x in the induced lattice ℒ(𝒜) = (A; ∨, ∧, (a)a∈A, 0, 1) if and only if it satisfies x ⊕ x = x.

Corollary 2.10

([20]). If a basic algebra 𝒜 = (A; ⊕, ¬, 0) satisfies the identity x ⊕ x = x, then the induced lattice ℒ(𝒜) is an ortholattice.

3 The Sheffer Stroke Reduction of Basic Algebras

As mentioned in [21], we can consider 𝒜 = (A; ⊕, ¬, 0) alternatively in signature {→, 0}, where x → y = ¬x ⊕ y. When regarding a logic axiomatized by a basic algebra, especially if 𝒜 is an MV-algebra, then it is a many-valued fuzzy logic, or if 𝒜 is an orthomodular lattice, it is a logic of quantum mechanics. Then the operation → plays the role of logic connective implication. There are also some interesting results related to implication operation used (see [22-27]). Starting from this point of view, we construct an alternative signature {|} which consists of only the Sheffer stroke operation. Hence, it is of some importance to investigate the Sheffer stroke reduct in a general setting for basic algebras. Now, we can define the following concept.

Definition 3.1

An algebra (A, |) of type (2) is called a Sheffer stroke basic algebra if the following identities hold:

(SH1) (x|(x|x))|(x|x) = x,

(SH2) (x|(y|y))|(y|y) = (y|(x|x))|(x|x),

(SH3) (((x|(y|y))|(y|y))|(z|z))|((x|(z|z))|(x|(z|z))) = x|(x|x).

First of all, we give some simple properties of Sheffer Stroke basic algebras. Then, we demonstrate that every such algebra has an algebraic constant 1 as is the case for implication basic algebras [20].

Lemma 3.2

Let (A, |) be a Sheffer Stroke basic algebra. Then there exists an algebraic constant element 1 ∈ A and (A, |) meets the following identities:

  1. x|(x|x) = 1,

  2. x|(1|1) = 1,

  3. 1|(x|x) = x,

  4. ((x|(y|y))|(y|y))|(y|y) = x|(y|y),

  5. (y|(x|(y|y)))|(x|(y|y)) = 1.

Proof

(i) : In (SH3), we substitute [z := y] and [y := x] simultaneously and we carry out (SH1). Then we obtain

x|(x|x)=(((x|(x|x))|(x|x))|(y|y))|((x|(y|y))|(x|(y|y)))=(x|(y|y))|((x|(y|y))|(x|(y|y))).

When we substitute [x := (x|(y|y))] in the above equation, we derive

(x|(y|y))|((x|(y|y))|(x|(y|y)))=((x|(y|y))|(y|y))|(((x|(y|y))|(y|y))|((x|(y|y))|(y|y))).

Therefore, we get

x|(x|x)=((x|(y|y))|(y|y))|(((x|(y|y))|(y|y))|((x|(y|y))|(y|y))).

By using (SH2), we deduce

x|(x|x)=((x|(y|y))|(y|y))|(((x|(y|y))|(y|y))|((x|(y|y))|(y|y)))=((y|(x|x))|(x|x))|(((y|(x|x))|(x|x))|((y|(x|x))|(x|x)))=y|(y|y).

Thus (A; |) satisfies the identity x|(x|x) = y|(y|y) for all x, y ∈ A. It means that (A; |) contains an algebraic constant which will be denoted by 1 and hence this system satisfies the identity x|(x|x) = 1.

(iii) : Applying (i) in (SH1), we obtain

x=(x|(x|x))|(x|x)=1|(x|x).

(υ) : By using (i) in (SH3), we get

(((x|(y|y))|(y|y))|(z|z))|(x|(z|z))=1.

Firstly, the substitution [x := y] and [y := x] gives

(((y|(x|x))|(x|x))|(z|z))|(y|(z|z))=1.

Substituting [x := (x|(y|y))] and [z := (x|(y|y))] in the latter equation yields

(((y|((x|(y|y))|(x|(y|y))))|((x|(y|y))|(x|(y|y))))|((x|(y|y))|(x|(y|y))))(y|((x|(y|y))|(x|(y|y))))=1.

By using (SH3) and the identity (iii), we conclude

y|((x|(y|y))|(x|(y|y)))=1.

(iv) : In (iii), the substitution [x := (x|(y|y))] implies

1|((x|(y|y))|(x|(y|y)))=x|(y|y).

And using (υ) for the value 1 in the above equation, we get

(y|((x|(y|y))|(x|(y|y))))|((x|(y|y))|(x|(y|y)))=x|(y|y).

By using (SH2), we obtain

((x|(y|y))|(y|y))|(y|y)=x|(y|y).

(ii) : In (υ), if we choose x = y then we get

x|((x|(x|x))|(x|(x|x)))=1.

From the identity (i), we obtain

x|(1|1)=1.

☐

Theorem 3.3

The axioms of Sheffer Stroke basic algebra are independent.

Proof

To prove this claim, we construct a model for each axiom in which that axiom is false while the others are true. Let 𝒜 = ({0, 1}, |𝒜) be our model defined in the following tables:

(1)Independence of (SH1):

We define the operation |𝒜 in the following table:

Table 1

Operation Table for Independence of (SH1)

|𝒜 0 1
0 0 0
1 0 0

Then |𝒜 satisfies (SH2) and (SH3), but not (SH1) since (1|(1|1))|(1|1) = 0 ≠ 1.

(2)Independence of (SH2):

Consider the operation |𝒜, defined as in the following table:

Table 2

Operation Table for Independence of (SH2)

|𝒜 0 1
0 0 1
1 1 0

Then |𝒜 satisfies (SH1) and (SH3), but not (SH2) because if we choose x = 1 and y = 0, then (1|(0|0))|(0|0) = 1 ≠ 0 = (0|(1|1))|(1|1).

(3)Independence of (SH3):

Define the operation |𝒜 as in Table 3:

Table 3

Operation Table for Independence of (SH3)

|𝒜 0 1
0 0 1
1 1 1

The model |𝒜 satisfies (SH1) and (SH2), but not (SH3). When we choose x = 0 and y = z = 1, we get (((0|(1|1))|(1|1))|(1|1))|((0|(1|1))|(0|(1|1))) = 1 ≠ 0 = 0|(0|0). ☐

To construct a bridge between basic algebras and Sheffer stroke basic algebras, we need the following theorem:

Theorem 3.4

Let 𝒜 = (A; ⊕, ¬, 0) be a basic algebra. We define x|y = ¬x ⊕ ¬y. Then (A; |) is a Sheffer Stroke basic algebra.

Proof

By using (BA1) − (BA4), Lemma 2.9 and Lemma 3.2, we can verify the identities (SH1) − (SH3) as follows:

(SH1)

(x|(x|x))|(x|x)=¬(¬x⊕¬(¬x⊕¬x))⊕¬(¬x⊕¬x)=¬(¬x⊕x)⊕x=0⊕x=1.

(SH2)

(x|(y|y))|(y|y)=¬(¬x⊕¬(¬y⊕¬y))⊕¬(¬y⊕¬y)=¬((¬y⊕¬y)⊕)⊕x=¬(¬y⊕x)⊕x,=¬(¬y⊕¬(¬x))⊕¬(¬x)=¬(¬y⊕¬(¬x⊕¬x))⊕¬(¬x⊕¬x)=(y|(x|x))|(x|x).

(SH3)

(((x|(y|y))|(y|y))|(z|z))|((x|(z|z))|(x|(z|z)))=¬(¬(¬(¬x⊕¬(¬y))⊕¬(¬z)))⊕¬(¬(¬x⊕¬(¬z))⊕¬(¬x⊕¬(¬z)))=¬(¬(¬(¬x⊕y)⊕y)⊕z)⊕¬(¬(¬x⊕z)⊕¬(¬x⊕z))=¬(¬(¬(¬x⊕y)⊕y)⊕z)⊕(¬x⊕z)=1=x|(x|x).

⃞

To reveal the structure of Sheffer stroke basic algebras, we introduce a partial order relation on A.

Lemma 3.5

Let (A; |) be a Sheffer stroke basic algebra. A binary relation ≤ is defined on A as follows:

x≤yifandonlyifx|(y|y)=1.

Then the binary relation ≤ is a partial order on A such that x ≤ 1 for each x ∈ A. Moreover, we have

z≤(x|(z|z))andx≤yimpliesy|(z|z)≤x|(z|z)

for all x, y, z ∈ A.

Proof

• Reflexivity follows from Lemma 3.2 (i).

• Assume that x ≤ y and y ≤ x. Then x|(y|y) = 1 and y|(x|x) = 1. From the hypothesis, (SH2) and Lemma 3.2 (iii), we obtain

x=1|(x|x)=(y|(x|x))|(x|x)=(x|(y|y))|(y|y)=1|(y|y)=y.

• Suppose that x ≤ y and y ≤ z. Then we have x|(y|y) = 1 and y|(z|z) = 1. Using this, (SH3) and Lemma 3.2 (iii), we get

1=(((x|(y|y))|(y|y))|(z|z))|((x|(z|z))|(x|(z|z)))=((1|(y|y))|(z|z))|((x|(z|z))|(x|(z|z)))=(y|(z|z))|((x|(z|z))|(x|(z|z)))=1|((x|(z|z))|(x|(z|z)))=x|(z|z).

Thus x ≤ z, showing that ≤ is a partial order on A. From Lemma 3.2 (ii), we get x ≤ 1 for each x ∈ A.

Moreover, assume that x ≤ y and z ∈ A. Then

1=(((x|(y|y))|(y|y))|(z|z))|((x|(z|z))|(x|(z|z)))=((1|(y|y))|(z|z))|((x|(z|z))|(x|(z|z)))=(y|(z|z))|((x|(z|z))|(x|(z|z)))

which means y|(z|z) ≤ x|(z|z). Putting here [y := 1], we obtain z = 1|(z|z) ≤ x|(z|z). ☐

The partial order ≤ introduced in the above lemma will be called induced partial order of the Sheffer stroke basic algebra (A; |).

Theorem 3.6

Let (A; |) be a Sheffer stroke basic algebra and ≤ its induced partial order. Then (A; ≤) is a join semi-lattice with the greatest element 1 where x ∨ y = (x|(y|y))|(y|y).

Proof

Using Lemma 3.5 and (SH2), we obtain x ≤ (x|(y|y))|(y|y) and y ≤ (y|(x|x))|(x|x) = (x|(y|y))|(y|y). Hence (x|(y|y))|(y|y) is an upper bound for x and y.

We assume x, y ≤ z. Then by using Lemma 3.5 twice we get

(x|(y|y))|(y|y)≤(z|(y|y))|(y|y)=(y|(z|z))|(z|z)=1|(z|z)=z.

Therefore, (x|(y|y))|(y|y) is the least upper bound for x and y, i.e., x ∨ y = (x|(y|y))|(y|y) is the supremum of x, y. ☐

Let (A; |) be a Sheffer stroke basic algebra. The semilattice (A; ∨) derived in the above theorem will be called the induced semilattice of (A; |).

Theorem 3.7

Let (A; |) be a Sheffer Stroke basic algebra and (A; ∨) its induced semi-lattice. For all p ∈ A, the closed interval [p, 1] is a lattice ([p, 1]; ∨, ∧p,p) with an antitone involution x ↦ xp where

xp=x|(p|p)andx∧py=((x|(p|p))∨(y|(p|p)))|(p|p)

for all x, y ∈ A.

Proof

Let x be in [p, 1]. Assume that x ≤ y. Then x|(y|y) = 1. Now substituting [z := p] into (SH3) we get

(((x|(y|y))|(y|y))|(p|p))|((x|(p|p))|(x|(p|p)))=1.

Since x|(y|y) = 1, by Lemma 3.2 (iii) we obtain

((y|(p|p))|((x|(p|p))|(x|(p|p)))=1.

By the definition of ≤, we get y|(p|p) ≤ x|(p|p), hence yp ≤ xp. So x ↦ xp is a partial order reversing mapping. In (SH3), substitute [y := 1] and [z := p]. Then

(((x|(1|1))|(1|1))|(p|p))|((x|(p|p))|(x|(p|p)))=1.

By Lemma 3.2 (ii) and (iii) we have

p|((x|(p|p))|(x|(p|p)))=1.

Hence p ≤ x|(p|p), i.e., p ≤ xp. Then x ↦ xp is a mapping of [p, 1] into itself. By Theorem 3.6, xpp = (x|(p|p))|(p|p) = x ∨ p = x and so it is an involution of [p, 1]. Then we can carry out De Morgan laws to show that

(xp∨yp)p=((x|(p|p))∨(y|(p|p)))p=((x|(p|p))∨(y|(p|p)))|(p|p)=x∧py.

This is the infimum of x,y ∈ [p, 1]. Consequently, ([p, 1]; ∨, ∧p,p ) is a lattice with an antitone involution. ☐

Corollary 3.8

Let (A; |) be a Sheffer Stroke basic algebra and ≤ is the induced partial order on this system. Then (A; ≤) is a join-semilattice which has the greatest element 1. For each p ∈ A the closed interval [p, 1] is a basic algebra ([p, 1]; ⊕p, ¬p, p) if x ⊕p y = (x|(p|p))|(y|y) and ¬p x = x|(p|p) are defined for all x, y ∈ A.

From now on, ([p, 1]; ⊕p, ¬p, p) is said to be an interval basic algebra with the greatest element 1 and the least element p. Therefore, Theorem 3.7 corresponds to the semilattice structure of a Sheffer stroke basic algebra. We prove that this explanation is complete, in other words, the other direction of Theorem 3.7 can be obtained.

Theorem 3.9

Let (A; ∨, 1) be a join-semilattice with the greatest element 1 such that for every p ∈ A, the closed interval [p, 1] is a lattice with antitone involution x ↦ xp. If we define x|y = (x ∨ yp)yp, then (A; |) is a Sheffer Stroke basic algebra.

We say that (A; |) is a Sheffer Stroke basic algebra which has the least element if there exists an element 0 ∈ A such that 0 ≤ a for all a ∈ A, where ≤ is the induced partial order. So, any Sheffer Stroke basic algebra with the least element 0 satisfies the identity 0|(x|x) = 1.

The proof of the following theorem is straightforward.

Theorem 3.10

Let (A; |) be a Sheffer Stroke basic algebra which has the least element 0. If we define ¬x = x|(0|0) and x ⊕ y = (x|(0|0))|(y|y), then the system (A; ⊕, ¬, 0) is a basic algebra and x|y = ¬x ⊕ ¬y.

In the remaining part of this work, we show that there is a bridge between Sheffer stroke basic algebras and Boolean algebras.

Lemma 3.11

Let (A; |) be a Sheffer stroke basic algebra and (A; ∨) its induced complemented semilattice. For each p ∈ A, the closed interval [p, 1] is a lattice ([p, 1]; ∨, ∧p ,p ) with an antitone involution. Then the following identities hold

x∨xp=1andx∧pxp=p

for each x ∈ [p, 1].

Proof

The definition of Sheffer stroke operation gives

(x|y)|(x|x)=x.

Substituting [y := (p|p)] into the latter equation yields

(x|(p|p))|(x|x)=x.

From the definition of ≤ we have

((x|(p|p))|(x|x))|(x|x)=1.

From (SH2) we get

(x|((x|(p|p))|(x|(p|p))))|((x|(p|p))|(x|(p|p)))=1.

So we can conclude that x ∨ (x|(p|p)) = 1 and x ∨ xp = 1. Now we have

x∧pxp=x∧p(x|(p|p))=((x|(p|p))∨((x|(p|p))|(p|p)))|(p|p)

Since p is the least element, we have x = x ∨ p = (x|(p|p))|(p|p). We use this equality in the latter equation and by Lemma 3.2 (iii) we obtain

(xp∨x)|(p|p)=1|(p|p)=p.

☐

Corollary 3.12

Let (A; |) be a Sheffer Stroke basic algebra with the least element 0 and the greatest element 1. Then (A; ∨, ∧0,0, 0, 1) is a lattice with an antitone involution.

Lemma 3.13

Let (A; |) be a Sheffer Stroke basic algebra with the least element 0 and the greatest element 1, and (A; ∨, ∧0,0, 0, 1) is a lattice with an antitone involution x ↦ x0. Then it satisfies the following properties for all x ∈ A:

  1. x|(0|0) = 0 if and only if x = 1,

  2. x|(k|k) = k if and only if x ∨ k = 1,

  3. 0|0 = 1 ,

  4. x|1 = x|x,

  5. x0 = x|x,

  6. (k|k)|(x|x) = 1 if and only if x ∨ k = 1.

Proof

  1. (⇒:) Assume that x|(0|0) = 0. Then (x|(0|0))|(0|0) = 1. Since x ↦ x0 is an involution, we have 1 = (x|(0|0))|(0|0) = x00 = x.

    (⇐:) It follows from Lemma 3.2 (iii).

  2. (⇒:)Let x|(k|k) = k. Then we have (x|(k|k))|(k|k) = 1; hence x ∨ k = 1.

    (⇐:) Assume that x ∨ k = 1. Then we have (x|(k|k))|(k|k) = 1. Now, we substitute [y := k] in Lemma 3.2 (iv). Thereafter by using the hypothesis and Lemma 3.2 (iii) we obtain

    x|(k|k)=((x|(k|k))|(k|k))|(k|k)=1|(k|k)=k.

  3. Assume that x ∈ A. Then 0 ≤ x and 0 ≤ (x|x). By Lemma 2.7 (iv), we obtain x|(x|x) ≤ 0|0 ≤ 1, hence 0|0 = 1.

  4. It follows from Lemma 2.6.

  5. By (iv), we have x0 = x|(0|0) = x|1 = x|x.

  6. (⇒:) In [28], we substitute [x := (k|k)] and [y := x] in the equation x|(y|y) = x|(x|y) to

    1=(k|k)|(x|x)=(k|k)|((k|k)|x)=x∨k.

    (⇐:) It is verified similarly. ☐

Theorem 3.14

Let (A; |) be a Sheffer Stroke basic algebra with the least element 0 and the greatest element 1, and (A; ∨, ∧0, 0, 0, 1) its induced complemented lattice with an antitone involution x ↦ x0. Then, there exists unique x0 such that

x∨x0=1andx∧0x0=0

for all x ∈ A.

Proof

We show that there exists x ↦ x0 an antitone involution of A such that x0 ∨ x = 1 and x0 ∧0 x = 0 in Lemma 3.11. For the uniqueness, assume that x0 = k and x0 = l. Then by Lemma 3.13 (vi), we have

k=x0=(x|x)andl=x0=(x|x).

Then from these equalities we get

x∨k=1⇒(x|x)|(k|k)=1⇒l|(k|k)=1⇒l≤k.

Using the same technique, we can obtain k ≤ l. Therefore, k = l. Hence, we have unique x0 such that x0 ∨ x = 1 for each x ∈ A. The identity x ∧0 x0 = 0 is verified similarly. ☐

Definition 3.15.

A Sheffer Stroke basic algebra (A; |) is commutative if it satisfies

(x|(p|p))|(y|y)=(y|(p|p))|(x|x)

for all x, y ∈ [p, 1].

Lemma 3.16

Let (A; |) be a commutative basic algebra. Then the interval basic algebra ([p, 1]; ⊕p, ¬p, p) is commutative for each p ∈ A.

From Theorem 2.8 in [20] we obtain the following corollary:

Corollary 3.17

Let (A; |) be a commutative Sheffer Stroke basic algebra and (A; ∨) its induced semilattice. Then

  1. the interval basic algebra ([p, 1]; ⊕p, ¬p, p) is commutative basic algebra for each p ∈ A,

  2. the interval lattice ([p, 1], ∨, ∧p) is distributive for each p ∈ A.

Theorem 3.18

Let (A; |) be a Sheffer Stroke basic algebra with the least element 0 and the greatest element 1, and (A; ∨, ∧0,0, 0, 1) its induced lattice with an antitone involution x ↦ x0. Then (A; ∨, ∧0,0, 0, 1) is Boolean algebra.

Proof

It follows fromTheorem 3.10 and Corollary 2.10. ☐

Acknowledgement

The authors thank the anonymous referees for his/her remarks which helped him to improve the presentation of the paper.

References

[1] Tarski A., Ein beitrag zur axiomatik der abelschen gruppen, Fundamenta Mathematicae, 1938, 30, 253-25610.4064/fm-30-1-253-256Suche in Google Scholar

[2] Sheffer H.M., A set of five independent postulates for Boolean algebras, with application to logical constants, Transactions of the American Mathematical Society, 1913, 14, 481-48810.1090/S0002-9947-1913-1500960-1Suche in Google Scholar

[3] McCune W., Verof R., Fitelson B., Harris K., Feist A., Wos L., Short single axioms for Boolean algebra, Journal of Automated Reasoning, 2002, 29, 1-1610.1023/A:1020542009983Suche in Google Scholar

[4] Abbott J.C., Semi-boolean algebra, Matematički Vesnik, 1967, 40, 177-198Suche in Google Scholar

[5] Abbott J.C., Implicational algebras, Bulletin mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie, 1967, 11,3-23Suche in Google Scholar

[6] Chajda I., Halaš R., An implication in orthologic, International Journal of Theoretical Physics, 2005, 44, 735-74410.1007/s10773-005-7051-1Suche in Google Scholar

[7] Chajda I., Orthomodular semilattices, Discrete mathematics, 2007, 307, 115-11810.1016/j.disc.2006.05.040Suche in Google Scholar

[8] Chajda I., Halaš R., Länger H., Orthomodular implication algebras, International Journal of Theoretical Physics, 2001, 40, 1875-188410.1023/A:1011933018776Suche in Google Scholar

[9] Chajda I., Emanovský P, Bounded lattices with antitone involutions and properties of MV-algebras, Discussiones Mathematicae, General Algebra and Applications, 2004, 24, 32-4210.7151/dmgaa.1073Suche in Google Scholar

[10] Chajda I., Lattices and semilattices having an antitone involution in every upper interval, Comment. Math. Univ. Carolin, 2003, 44, 577-585Suche in Google Scholar

[11] Chajda I., Kühr J., Basic algebras, Clone Theory and Discrete Mathematics Algebra and Logic Related to Computer Science, 2013Suche in Google Scholar

[12] Chajda I., Kolařík M., Independence of axiom system of basic algebras, Soft Computing, 2009, 13, 41-4310.1007/s00500-008-0291-2Suche in Google Scholar

[13] Chajda I., Kolařík M., Interval basic algebras, Novi Sad J. Math, 2009, 39, 71-78Suche in Google Scholar

[14] Cignoli R. L., d’Ottaviano I. M. , Mundici D., Algebraic foundations of many-valued reasoning, Springer Science and Business Media, 201310.1007/978-94-015-9480-6Suche in Google Scholar

[15] Chajda I., Basic algebras and their applications, an overview, Proceedings of the Salzburg Conference, Verlag Johannes Heyn, Klagenfurt, 2011Suche in Google Scholar

[16] Chajda I., Halaš R., Kühr J., Many-valued quantum algebras, Algebra Universalis, 2009, 60, 63-9010.1007/s00012-008-2086-9Suche in Google Scholar

[17] Birkhoff G., Lattice Theory, Proc. Amer. Math. Soc., Providence, R. I. third edition, 196710.1090/coll/025Suche in Google Scholar

[18] Grätzer G., General Lattice Theory, Springer Science and Business Media, 2002Suche in Google Scholar

[19] Chajda I., Sheffer operation in ortholattices, Acta Universitatis Palackianae Olomucensis, Facultas Rerum Naturalium Mathematica, 2005, 44, 19-23Suche in Google Scholar

[20] Chajda I., Basic algebras, logics, trends and applications, Asian-European Journal of Mathematics, 2015, 8, (46 pages)10.1142/S1793557115500400Suche in Google Scholar

[21] Chajda I., Kolařík M., Švrček F., Implication and equivalential reducts of basic algebras, Acta Universitatis Palackianae Olomucensis Facultas Rerum Naturalium Mathematica, 2010, 49, 21-36Suche in Google Scholar

[22] Chajda I., Halaš R., Länger H., Simple axioms for orthomodular implication algebras, International Journal of Theoretical Physics, 2004, 43, 911-91410.1023/B:IJTP.0000048587.50827.93Suche in Google Scholar

[23] Chajda I., Filters of implication reducts of basic algebras, Soft Computing, 2014, 18, 849-85210.1007/s00500-013-1200-xSuche in Google Scholar

[24] Chajda I., Halaš R., Kühr J., Semilattice structures, Heldermann, Michigan University, 2007Suche in Google Scholar

[25] Piziak R., Orthomodular lattices as implication algebras, Journal of Philosophical Logic, 1974, 3, 413-41810.1007/BF00257483Suche in Google Scholar

[26] Pavicic M., Megill N. D., Quantum and classical implication algebras with primitive implications, International Journal of Theoretical Physics, 1998, 37, 2091-209810.1023/A:1026685801864Suche in Google Scholar

[27] Herman L., Marsden E. L., Piziak R., Implication connectives in orthomodular lattices, Notre Dame Journal of Formal Logic, 1975, 16, 305-32810.1305/ndjfl/1093891789Suche in Google Scholar

[28] Łukaszuk A., Grabowski A., Short Sheffer Stroke-Based Single Axiom for Boolean Algebras, Formalized Mathematics, 1975, 12, 363-370Suche in Google Scholar
Received: 2017-4-4
Accepted: 2017-5-22
Published Online: 2017-7-13

© 2017 Oner and Senturk

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Artikel in diesem Heft

  1. Regular Articles
  2. Integrals of Frullani type and the method of brackets
  3. Regular Articles
  4. Edge of chaos in reaction diffusion CNN model
  5. Regular Articles
  6. Calculus using proximities: a mathematical approach in which students can actually prove theorems
  7. Regular Articles
  8. An investigation on hyper S-posets over ordered semihypergroups
  9. Regular Articles
  10. The Leibniz algebras whose subalgebras are ideals
  11. Regular Articles
  12. Fixed point and multidimensional fixed point theorems with applications to nonlinear matrix equations in terms of weak altering distance functions
  13. Regular Articles
  14. Matrix rank and inertia formulas in the analysis of general linear models
  15. Regular Articles
  16. The hybrid power mean of quartic Gauss sums and Kloosterman sums
  17. Regular Articles
  18. Tauberian theorems for statistically (C,1,1) summable double sequences of fuzzy numbers
  19. Regular Articles
  20. Some properties of graded comultiplication modules
  21. Regular Articles
  22. The characterizations of upper approximation operators based on special coverings
  23. Regular Articles
  24. Bi-integrable and tri-integrable couplings of a soliton hierarchy associated with SO(4)
  25. Regular Articles
  26. Dynamics for a discrete competition and cooperation model of two enterprises with multiple delays and feedback controls
  27. Regular Articles
  28. A new view of relationship between atomic posets and complete (algebraic) lattices
  29. Regular Articles
  30. A class of extensions of Restricted (s, t)-Wythoff’s game
  31. Regular Articles
  32. New bounds for the minimum eigenvalue of 𝓜-tensors
  33. Regular Articles
  34. Shintani and Shimura lifts of cusp forms on certain arithmetic groups and their applications
  35. Regular Articles
  36. Empirical likelihood for quantile regression models with response data missing at random
  37. Regular Articles
  38. Convex combination of analytic functions
  39. Regular Articles
  40. On the Yang-Baxter-like matrix equation for rank-two matrices
  41. Regular Articles
  42. Uniform topology on EQ-algebras
  43. Regular Articles
  44. Integrations on rings
  45. Regular Articles
  46. The quasilinear parabolic kirchhoff equation
  47. Regular Articles
  48. Avoiding rainbow 2-connected subgraphs
  49. Regular Articles
  50. On non-Hopfian groups of fractions
  51. Regular Articles
  52. Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions
  53. Regular Articles
  54. Rings in which elements are the sum of a nilpotent and a root of a fixed polynomial that commute
  55. Regular Articles
  56. Superstability of functional equations related to spherical functions
  57. Regular Articles
  58. Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52
  59. Regular Articles
  60. Weighted minimal translation surfaces in the Galilean space with density
  61. Regular Articles
  62. Complete convergence for weighted sums of pairwise independent random variables
  63. Regular Articles
  64. Binomials transformation formulae for scaled Fibonacci numbers
  65. Regular Articles
  66. Growth functions for some uniformly amenable groups
  67. Regular Articles
  68. Hopf bifurcations in a three-species food chain system with multiple delays
  69. Regular Articles
  70. Oscillation and nonoscillation of half-linear Euler type differential equations with different periodic coefficients
  71. Regular Articles
  72. Osculating curves in 4-dimensional semi-Euclidean space with index 2
  73. Regular Articles
  74. Some new facts about group 𝒢 generated by the family of convergent permutations
  75. Regular Articles
  76. lnfinitely many solutions for fractional Schrödinger equations with perturbation via variational methods
  77. Regular Articles
  78. Supersolvable orders and inductively free arrangements
  79. Regular Articles
  80. Asymptotically almost automorphic solutions of differential equations with piecewise constant argument
  81. Regular Articles
  82. Finite groups whose all second maximal subgroups are cyclic
  83. Regular Articles
  84. Semilinear systems with a multi-valued nonlinear term
  85. Regular Articles
  86. Positive solutions for Hadamard differential systems with fractional integral conditions on an unbounded domain
  87. Regular Articles
  88. Calibration and simulation of Heston model
  89. Regular Articles
  90. One kind sixth power mean of the three-term exponential sums
  91. Regular Articles
  92. Cyclic pairs and common best proximity points in uniformly convex Banach spaces
  93. Regular Articles
  94. The uniqueness of meromorphic functions in k-punctured complex plane
  95. Regular Articles
  96. Normalizers of intermediate congruence subgroups of the Hecke subgroups
  97. Regular Articles
  98. The hyperbolicity constant of infinite circulant graphs
  99. Regular Articles
  100. Scott convergence and fuzzy Scott topology on L-posets
  101. Regular Articles
  102. One sided strong laws for random variables with infinite mean
  103. Regular Articles
  104. The join of split graphs whose completely regular endomorphisms form a monoid
  105. Regular Articles
  106. A new branch and bound algorithm for minimax ratios problems
  107. Regular Articles
  108. Upper bound estimate of incomplete Cochrane sum
  109. Regular Articles
  110. Value distributions of solutions to complex linear differential equations in angular domains
  111. Regular Articles
  112. The nonlinear diffusion equation of the ideal barotropic gas through a porous medium
  113. Regular Articles
  114. The Sheffer stroke operation reducts of basic algebras
  115. Regular Articles
  116. Extensions and improvements of Sherman’s and related inequalities for n-convex functions
  117. Regular Articles
  118. Classification lattices are geometric for complete atomistic lattices
  119. Regular Articles
  120. Possible numbers of x’s in an {x, y}-matrix with a given rank
  121. Regular Articles
  122. New error bounds for linear complementarity problems of weakly chained diagonally dominant B-matrices
  123. Regular Articles
  124. Boundedness of vector-valued B-singular integral operators in Lebesgue spaces
  125. Regular Articles
  126. On the Golomb’s conjecture and Lehmer’s numbers
  127. Regular Articles
  128. Some applications of the Archimedean copulas in the proof of the almost sure central limit theorem for ordinary maxima
  129. Regular Articles
  130. Dual-stage adaptive finite-time modified function projective multi-lag combined synchronization for multiple uncertain chaotic systems
  131. Regular Articles
  132. Corrigendum to: Dual-stage adaptive finite-time modified function projective multi-lag combined synchronization for multiple uncertain chaotic systems
  133. Regular Articles
  134. Convergence and stability of generalized φ-weak contraction mapping in CAT(0) spaces
  135. Regular Articles
  136. Triple solutions for a Dirichlet boundary value problem involving a perturbed discrete p(k)-Laplacian operator
  137. Regular Articles
  138. OD-characterization of alternating groups Ap+d
  139. Regular Articles
  140. On Jordan mappings of inverse semirings
  141. Regular Articles
  142. On generalized Ehresmann semigroups
  143. Regular Articles
  144. On topological properties of spaces obtained by the double band matrix
  145. Regular Articles
  146. Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions
  147. Regular Articles
  148. Chain conditions on composite Hurwitz series rings
  149. Regular Articles
  150. Coloring subgraphs with restricted amounts of hues
  151. Regular Articles
  152. An extension of the method of brackets. Part 1
  153. Regular Articles
  154. Branch-delete-bound algorithm for globally solving quadratically constrained quadratic programs
  155. Regular Articles
  156. Strong edge geodetic problem in networks
  157. Regular Articles
  158. Ricci solitons on almost Kenmotsu 3-manifolds
  159. Regular Articles
  160. Uniqueness of meromorphic functions sharing two finite sets
  161. Regular Articles
  162. On the fourth-order linear recurrence formula related to classical Gauss sums
  163. Regular Articles
  164. Dynamical behavior for a stochastic two-species competitive model
  165. Regular Articles
  166. Two new eigenvalue localization sets for tensors and theirs applications
  167. Regular Articles
  168. κ-strong sequences and the existence of generalized independent families
  169. Regular Articles
  170. Commutators of Littlewood-Paley gκ∗ -functions on non-homogeneous metric measure spaces
  171. Regular Articles
  172. On decompositions of estimators under a general linear model with partial parameter restrictions
  173. Regular Articles
  174. Groups and monoids of Pythagorean triples connected to conics
  175. Regular Articles
  176. Hom-Lie superalgebra structures on exceptional simple Lie superalgebras of vector fields
  177. Regular Articles
  178. Numerical methods for the multiplicative partial differential equations
  179. Regular Articles
  180. Solvable Leibniz algebras with NFn⊕ Fm1 nilradical
  181. Regular Articles
  182. Evaluation of the convolution sums ∑al+bm=n lσ(l) σ(m) with ab ≤ 9
  183. Regular Articles
  184. A study on soft rough semigroups and corresponding decision making applications
  185. Regular Articles
  186. Some new inequalities of Hermite-Hadamard type for s-convex functions with applications
  187. Regular Articles
  188. Deficiency of forests
  189. Regular Articles
  190. Perfect codes in power graphs of finite groups
  191. Regular Articles
  192. A new compact finite difference quasilinearization method for nonlinear evolution partial differential equations
  193. Regular Articles
  194. Does any convex quadrilateral have circumscribed ellipses?
  195. Regular Articles
  196. The dynamic of a Lie group endomorphism
  197. Regular Articles
  198. On pairs of equations in unlike powers of primes and powers of 2
  199. Regular Articles
  200. Differential subordination and convexity criteria of integral operators
  201. Regular Articles
  202. Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem
  203. Regular Articles
  204. On θ-commutators and the corresponding non-commuting graphs
  205. Regular Articles
  206. Quasi-maximum likelihood estimator of Laplace (1, 1) for GARCH models
  207. Regular Articles
  208. Multiple and sign-changing solutions for discrete Robin boundary value problem with parameter dependence
  209. Regular Articles
  210. Fundamental relation on m-idempotent hyperrings
  211. Regular Articles
  212. A novel recursive method to reconstruct multivariate functions on the unit cube
  213. Regular Articles
  214. Nabla inequalities and permanence for a logistic integrodifferential equation on time scales
  215. Regular Articles
  216. Enumeration of spanning trees in the sequence of Dürer graphs
  217. Regular Articles
  218. Quotient of information matrices in comparison of linear experiments for quadratic estimation
  219. Regular Articles
  220. Fourier series of functions involving higher-order ordered Bell polynomials
  221. Regular Articles
  222. Simple modules over Auslander regular rings
  223. Regular Articles
  224. Weighted multilinear p-adic Hardy operators and commutators
  225. Regular Articles
  226. Guaranteed cost finite-time control of positive switched nonlinear systems with D-perturbation
  227. Regular Articles
  228. A modified quasi-boundary value method for an abstract ill-posed biparabolic problem
  229. Regular Articles
  230. Extended Riemann-Liouville type fractional derivative operator with applications
  231. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  232. The algebraic size of the family of injective operators
  233. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  234. The history of a general criterium on spaceability
  235. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  236. On sequences not enjoying Schur’s property
  237. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  238. A hierarchy in the family of real surjective functions
  239. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  240. Dynamics of multivalued linear operators
  241. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  242. Linear dynamics of semigroups generated by differential operators
  243. Special Issue on Recent Developments in Differential Equations
  244. Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces
  245. Special Issue on Recent Developments in Differential Equations
  246. Determination of a diffusion coefficient in a quasilinear parabolic equation
  247. Special Issue on Recent Developments in Differential Equations
  248. Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables
  249. Special Issue on Recent Developments in Differential Equations
  250. A nonlinear plate control without linearization
  251. Special Issue on Recent Developments in Differential Equations
  252. Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations
  253. Special Issue on Recent Developments in Differential Equations
  254. Inverse problem for a physiologically structured population model with variable-effort harvesting
  255. Special Issue on Recent Developments in Differential Equations
  256. Existence of solutions for delay evolution equations with nonlocal conditions
  257. Special Issue on Recent Developments in Differential Equations
  258. Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities
  259. Special Issue on Recent Developments in Differential Equations
  260. Coupled fixed point theorems in complete metric spaces endowed with a directed graph and application
  261. Special Issue on Recent Developments in Differential Equations
  262. Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations
  263. Special Issue on Recent Developments in Differential Equations
  264. Integro-differential systems with variable exponents of nonlinearity
  265. Special Issue on Recent Developments in Differential Equations
  266. Elliptic operators on refined Sobolev scales on vector bundles
  267. Special Issue on Recent Developments in Differential Equations
  268. Multiplicity solutions of a class fractional Schrödinger equations
  269. Special Issue on Recent Developments in Differential Equations
  270. Determining of right-hand side of higher order ultraparabolic equation
  271. Special Issue on Recent Developments in Differential Equations
  272. Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain
  273. Topical Issue on Metaheuristics - Methods and Applications
  274. Learnheuristics: hybridizing metaheuristics with machine learning for optimization with dynamic inputs
  275. Topical Issue on Metaheuristics - Methods and Applications
  276. Nature–inspired metaheuristic algorithms to find near–OGR sequences for WDM channel allocation and their performance comparison
  277. Topical Issue on Cyber-security Mathematics
  278. Monomial codes seen as invariant subspaces
  279. Topical Issue on Cyber-security Mathematics
  280. Expert knowledge and data analysis for detecting advanced persistent threats
  281. Topical Issue on Cyber-security Mathematics
  282. Feedback equivalence of convolutional codes over finite rings
https://doi.org/10.1515/math-2017-0075
Schlagwörter für diesen Artikel
Basic algebras; Sheffer stroke reduct; Lattice with antitone involutions
Creative Commons
BY-NC-ND 3.0

Artikel in diesem Heft

  1. Regular Articles
  2. Integrals of Frullani type and the method of brackets
  3. Regular Articles
  4. Edge of chaos in reaction diffusion CNN model
  5. Regular Articles
  6. Calculus using proximities: a mathematical approach in which students can actually prove theorems
  7. Regular Articles
  8. An investigation on hyper S-posets over ordered semihypergroups
  9. Regular Articles
  10. The Leibniz algebras whose subalgebras are ideals
  11. Regular Articles
  12. Fixed point and multidimensional fixed point theorems with applications to nonlinear matrix equations in terms of weak altering distance functions
  13. Regular Articles
  14. Matrix rank and inertia formulas in the analysis of general linear models
  15. Regular Articles
  16. The hybrid power mean of quartic Gauss sums and Kloosterman sums
  17. Regular Articles
  18. Tauberian theorems for statistically (C,1,1) summable double sequences of fuzzy numbers
  19. Regular Articles
  20. Some properties of graded comultiplication modules
  21. Regular Articles
  22. The characterizations of upper approximation operators based on special coverings
  23. Regular Articles
  24. Bi-integrable and tri-integrable couplings of a soliton hierarchy associated with SO(4)
  25. Regular Articles
  26. Dynamics for a discrete competition and cooperation model of two enterprises with multiple delays and feedback controls
  27. Regular Articles
  28. A new view of relationship between atomic posets and complete (algebraic) lattices
  29. Regular Articles
  30. A class of extensions of Restricted (s, t)-Wythoff’s game
  31. Regular Articles
  32. New bounds for the minimum eigenvalue of 𝓜-tensors
  33. Regular Articles
  34. Shintani and Shimura lifts of cusp forms on certain arithmetic groups and their applications
  35. Regular Articles
  36. Empirical likelihood for quantile regression models with response data missing at random
  37. Regular Articles
  38. Convex combination of analytic functions
  39. Regular Articles
  40. On the Yang-Baxter-like matrix equation for rank-two matrices
  41. Regular Articles
  42. Uniform topology on EQ-algebras
  43. Regular Articles
  44. Integrations on rings
  45. Regular Articles
  46. The quasilinear parabolic kirchhoff equation
  47. Regular Articles
  48. Avoiding rainbow 2-connected subgraphs
  49. Regular Articles
  50. On non-Hopfian groups of fractions
  51. Regular Articles
  52. Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions
  53. Regular Articles
  54. Rings in which elements are the sum of a nilpotent and a root of a fixed polynomial that commute
  55. Regular Articles
  56. Superstability of functional equations related to spherical functions
  57. Regular Articles
  58. Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52
  59. Regular Articles
  60. Weighted minimal translation surfaces in the Galilean space with density
  61. Regular Articles
  62. Complete convergence for weighted sums of pairwise independent random variables
  63. Regular Articles
  64. Binomials transformation formulae for scaled Fibonacci numbers
  65. Regular Articles
  66. Growth functions for some uniformly amenable groups
  67. Regular Articles
  68. Hopf bifurcations in a three-species food chain system with multiple delays
  69. Regular Articles
  70. Oscillation and nonoscillation of half-linear Euler type differential equations with different periodic coefficients
  71. Regular Articles
  72. Osculating curves in 4-dimensional semi-Euclidean space with index 2
  73. Regular Articles
  74. Some new facts about group 𝒢 generated by the family of convergent permutations
  75. Regular Articles
  76. lnfinitely many solutions for fractional Schrödinger equations with perturbation via variational methods
  77. Regular Articles
  78. Supersolvable orders and inductively free arrangements
  79. Regular Articles
  80. Asymptotically almost automorphic solutions of differential equations with piecewise constant argument
  81. Regular Articles
  82. Finite groups whose all second maximal subgroups are cyclic
  83. Regular Articles
  84. Semilinear systems with a multi-valued nonlinear term
  85. Regular Articles
  86. Positive solutions for Hadamard differential systems with fractional integral conditions on an unbounded domain
  87. Regular Articles
  88. Calibration and simulation of Heston model
  89. Regular Articles
  90. One kind sixth power mean of the three-term exponential sums
  91. Regular Articles
  92. Cyclic pairs and common best proximity points in uniformly convex Banach spaces
  93. Regular Articles
  94. The uniqueness of meromorphic functions in k-punctured complex plane
  95. Regular Articles
  96. Normalizers of intermediate congruence subgroups of the Hecke subgroups
  97. Regular Articles
  98. The hyperbolicity constant of infinite circulant graphs
  99. Regular Articles
  100. Scott convergence and fuzzy Scott topology on L-posets
  101. Regular Articles
  102. One sided strong laws for random variables with infinite mean
  103. Regular Articles
  104. The join of split graphs whose completely regular endomorphisms form a monoid
  105. Regular Articles
  106. A new branch and bound algorithm for minimax ratios problems
  107. Regular Articles
  108. Upper bound estimate of incomplete Cochrane sum
  109. Regular Articles
  110. Value distributions of solutions to complex linear differential equations in angular domains
  111. Regular Articles
  112. The nonlinear diffusion equation of the ideal barotropic gas through a porous medium
  113. Regular Articles
  114. The Sheffer stroke operation reducts of basic algebras
  115. Regular Articles
  116. Extensions and improvements of Sherman’s and related inequalities for n-convex functions
  117. Regular Articles
  118. Classification lattices are geometric for complete atomistic lattices
  119. Regular Articles
  120. Possible numbers of x’s in an {x, y}-matrix with a given rank
  121. Regular Articles
  122. New error bounds for linear complementarity problems of weakly chained diagonally dominant B-matrices
  123. Regular Articles
  124. Boundedness of vector-valued B-singular integral operators in Lebesgue spaces
  125. Regular Articles
  126. On the Golomb’s conjecture and Lehmer’s numbers
  127. Regular Articles
  128. Some applications of the Archimedean copulas in the proof of the almost sure central limit theorem for ordinary maxima
  129. Regular Articles
  130. Dual-stage adaptive finite-time modified function projective multi-lag combined synchronization for multiple uncertain chaotic systems
  131. Regular Articles
  132. Corrigendum to: Dual-stage adaptive finite-time modified function projective multi-lag combined synchronization for multiple uncertain chaotic systems
  133. Regular Articles
  134. Convergence and stability of generalized φ-weak contraction mapping in CAT(0) spaces
  135. Regular Articles
  136. Triple solutions for a Dirichlet boundary value problem involving a perturbed discrete p(k)-Laplacian operator
  137. Regular Articles
  138. OD-characterization of alternating groups Ap+d
  139. Regular Articles
  140. On Jordan mappings of inverse semirings
  141. Regular Articles
  142. On generalized Ehresmann semigroups
  143. Regular Articles
  144. On topological properties of spaces obtained by the double band matrix
  145. Regular Articles
  146. Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions
  147. Regular Articles
  148. Chain conditions on composite Hurwitz series rings
  149. Regular Articles
  150. Coloring subgraphs with restricted amounts of hues
  151. Regular Articles
  152. An extension of the method of brackets. Part 1
  153. Regular Articles
  154. Branch-delete-bound algorithm for globally solving quadratically constrained quadratic programs
  155. Regular Articles
  156. Strong edge geodetic problem in networks
  157. Regular Articles
  158. Ricci solitons on almost Kenmotsu 3-manifolds
  159. Regular Articles
  160. Uniqueness of meromorphic functions sharing two finite sets
  161. Regular Articles
  162. On the fourth-order linear recurrence formula related to classical Gauss sums
  163. Regular Articles
  164. Dynamical behavior for a stochastic two-species competitive model
  165. Regular Articles
  166. Two new eigenvalue localization sets for tensors and theirs applications
  167. Regular Articles
  168. κ-strong sequences and the existence of generalized independent families
  169. Regular Articles
  170. Commutators of Littlewood-Paley gκ∗ -functions on non-homogeneous metric measure spaces
  171. Regular Articles
  172. On decompositions of estimators under a general linear model with partial parameter restrictions
  173. Regular Articles
  174. Groups and monoids of Pythagorean triples connected to conics
  175. Regular Articles
  176. Hom-Lie superalgebra structures on exceptional simple Lie superalgebras of vector fields
  177. Regular Articles
  178. Numerical methods for the multiplicative partial differential equations
  179. Regular Articles
  180. Solvable Leibniz algebras with NFn⊕ Fm1 nilradical
  181. Regular Articles
  182. Evaluation of the convolution sums ∑al+bm=n lσ(l) σ(m) with ab ≤ 9
  183. Regular Articles
  184. A study on soft rough semigroups and corresponding decision making applications
  185. Regular Articles
  186. Some new inequalities of Hermite-Hadamard type for s-convex functions with applications
  187. Regular Articles
  188. Deficiency of forests
  189. Regular Articles
  190. Perfect codes in power graphs of finite groups
  191. Regular Articles
  192. A new compact finite difference quasilinearization method for nonlinear evolution partial differential equations
  193. Regular Articles
  194. Does any convex quadrilateral have circumscribed ellipses?
  195. Regular Articles
  196. The dynamic of a Lie group endomorphism
  197. Regular Articles
  198. On pairs of equations in unlike powers of primes and powers of 2
  199. Regular Articles
  200. Differential subordination and convexity criteria of integral operators
  201. Regular Articles
  202. Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem
  203. Regular Articles
  204. On θ-commutators and the corresponding non-commuting graphs
  205. Regular Articles
  206. Quasi-maximum likelihood estimator of Laplace (1, 1) for GARCH models
  207. Regular Articles
  208. Multiple and sign-changing solutions for discrete Robin boundary value problem with parameter dependence
  209. Regular Articles
  210. Fundamental relation on m-idempotent hyperrings
  211. Regular Articles
  212. A novel recursive method to reconstruct multivariate functions on the unit cube
  213. Regular Articles
  214. Nabla inequalities and permanence for a logistic integrodifferential equation on time scales
  215. Regular Articles
  216. Enumeration of spanning trees in the sequence of Dürer graphs
  217. Regular Articles
  218. Quotient of information matrices in comparison of linear experiments for quadratic estimation
  219. Regular Articles
  220. Fourier series of functions involving higher-order ordered Bell polynomials
  221. Regular Articles
  222. Simple modules over Auslander regular rings
  223. Regular Articles
  224. Weighted multilinear p-adic Hardy operators and commutators
  225. Regular Articles
  226. Guaranteed cost finite-time control of positive switched nonlinear systems with D-perturbation
  227. Regular Articles
  228. A modified quasi-boundary value method for an abstract ill-posed biparabolic problem
  229. Regular Articles
  230. Extended Riemann-Liouville type fractional derivative operator with applications
  231. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  232. The algebraic size of the family of injective operators
  233. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  234. The history of a general criterium on spaceability
  235. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  236. On sequences not enjoying Schur’s property
  237. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  238. A hierarchy in the family of real surjective functions
  239. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  240. Dynamics of multivalued linear operators
  241. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  242. Linear dynamics of semigroups generated by differential operators
  243. Special Issue on Recent Developments in Differential Equations
  244. Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces
  245. Special Issue on Recent Developments in Differential Equations
  246. Determination of a diffusion coefficient in a quasilinear parabolic equation
  247. Special Issue on Recent Developments in Differential Equations
  248. Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables
  249. Special Issue on Recent Developments in Differential Equations
  250. A nonlinear plate control without linearization
  251. Special Issue on Recent Developments in Differential Equations
  252. Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations
  253. Special Issue on Recent Developments in Differential Equations
  254. Inverse problem for a physiologically structured population model with variable-effort harvesting
  255. Special Issue on Recent Developments in Differential Equations
  256. Existence of solutions for delay evolution equations with nonlocal conditions
  257. Special Issue on Recent Developments in Differential Equations
  258. Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities
  259. Special Issue on Recent Developments in Differential Equations
  260. Coupled fixed point theorems in complete metric spaces endowed with a directed graph and application
  261. Special Issue on Recent Developments in Differential Equations
  262. Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations
  263. Special Issue on Recent Developments in Differential Equations
  264. Integro-differential systems with variable exponents of nonlinearity
  265. Special Issue on Recent Developments in Differential Equations
  266. Elliptic operators on refined Sobolev scales on vector bundles
  267. Special Issue on Recent Developments in Differential Equations
  268. Multiplicity solutions of a class fractional Schrödinger equations
  269. Special Issue on Recent Developments in Differential Equations
  270. Determining of right-hand side of higher order ultraparabolic equation
  271. Special Issue on Recent Developments in Differential Equations
  272. Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain
  273. Topical Issue on Metaheuristics - Methods and Applications
  274. Learnheuristics: hybridizing metaheuristics with machine learning for optimization with dynamic inputs
  275. Topical Issue on Metaheuristics - Methods and Applications
  276. Nature–inspired metaheuristic algorithms to find near–OGR sequences for WDM channel allocation and their performance comparison
  277. Topical Issue on Cyber-security Mathematics
  278. Monomial codes seen as invariant subspaces
  279. Topical Issue on Cyber-security Mathematics
  280. Expert knowledge and data analysis for detecting advanced persistent threats
  281. Topical Issue on Cyber-security Mathematics
  282. Feedback equivalence of convolutional codes over finite rings
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 16.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/math-2017-0075/html
Button zum nach oben scrollen