This document discusses derivations on lattices. It begins with background definitions of lattices, distributive lattices, modular lattices, and derivations. It then introduces the notion of f(x⋀y) = x⋀fy for a derivation f on a lattice L. It establishes some equivalence relations using isotone derivations and extends previous results on isotone derivations for distributive lattices. Finally, it shows that the set of all isotone derivations on a modular lattice forms a modular lattice itself under the operations of meet and join.
- The document discusses Fourier series and integrals.
- Fourier series decomposes a periodic function into a sum of sines and cosines. It is useful for representing periodic and discontinuous functions.
- There are three types of Fourier integrals: the general Fourier integral, Fourier cosine integral, and Fourier sine integral. These are used to represent functions over infinite intervals.
The document is about difference equations and includes:
1) An introduction to difference equations, what they are, and their objectives.
2) Examples of testing solutions by plugging them into difference equations.
3) A "guess and check" method for finding the terms of a sequence defined by a difference equation.
[2017년 SW 마에스트로 100+ 컨퍼런스]
- 발표자: 오픈스택 한국 커뮤니티 조성수
- 행사 정보: https://www.facebook.com/swmaestro/photos/a.816861878341341.1073741828.812223648805164/1832957773398408/?type=3&theater&ifg=1
This document introduces tensors through examples. It defines a vector as a rank 1 tensor and a matrix as a rank 2 tensor. It then provides an example of a rank 3 tensor. The document discusses how to define an inner product between tensors and provides examples using vectors and matrices. It also gives an example of how derivatives of a function can produce tensors of different ranks. Finally, it introduces the concept of decomposing matrices into their symmetric and antisymmetric parts.
The document defines ordered pairs, product sets, relations, and digraphs. It provides examples of defining relations between sets and representing them using matrices and digraphs. It introduces concepts such as the domain and range of a relation. It also describes paths in relations and digraphs, and how to compute higher powers of the relation matrix to determine connectivity between elements.
- The document discusses Fourier series and integrals.
- Fourier series decomposes a periodic function into a sum of sines and cosines. It is useful for representing periodic and discontinuous functions.
- There are three types of Fourier integrals: the general Fourier integral, Fourier cosine integral, and Fourier sine integral. These are used to represent functions over infinite intervals.
The document is about difference equations and includes:
1) An introduction to difference equations, what they are, and their objectives.
2) Examples of testing solutions by plugging them into difference equations.
3) A "guess and check" method for finding the terms of a sequence defined by a difference equation.
[2017년 SW 마에스트로 100+ 컨퍼런스]
- 발표자: 오픈스택 한국 커뮤니티 조성수
- 행사 정보: https://www.facebook.com/swmaestro/photos/a.816861878341341.1073741828.812223648805164/1832957773398408/?type=3&theater&ifg=1
This document introduces tensors through examples. It defines a vector as a rank 1 tensor and a matrix as a rank 2 tensor. It then provides an example of a rank 3 tensor. The document discusses how to define an inner product between tensors and provides examples using vectors and matrices. It also gives an example of how derivatives of a function can produce tensors of different ranks. Finally, it introduces the concept of decomposing matrices into their symmetric and antisymmetric parts.
The document defines ordered pairs, product sets, relations, and digraphs. It provides examples of defining relations between sets and representing them using matrices and digraphs. It introduces concepts such as the domain and range of a relation. It also describes paths in relations and digraphs, and how to compute higher powers of the relation matrix to determine connectivity between elements.
This document provides an overview of relations and their properties in discrete mathematics. It defines what a relation is, distinguishes between relations and functions, and describes key properties of relations including:
- Reflexive relations, where every element is related to itself.
- Symmetric relations, where if a is related to b then b is related to a.
- Transitive relations, where if a is related to b and b is related to c, then a is related to c.
It also discusses how to determine if a relation has these properties, combines multiple relations using set operations, and defines the composite of two relations. The overall goal is for students to understand relations and be able to analyze them for
This document discusses inner product spaces and how inner products can be defined on vector spaces to generalize concepts like the dot product, vector norms, angles between vectors, and distances between vectors. It provides examples of defining inner products on spaces like Rn, the space of polynomials Pn, and the space of 2x2 matrices M22. It shows how norms, orthogonality, and distances can be calculated in these spaces based on their defined inner products. The document also discusses how different inner products can lead to different geometries beyond standard Euclidean geometry.
This document discusses LU decomposition, which is a matrix factorization technique used to solve systems of linear equations. It breaks a matrix down into the product of a lower triangular matrix and an upper triangular matrix. This allows the system of equations to be solved through back substitution and forward substitution. The document also covers LDU, LUP, and LU decompositions with full pivoting, which are variations of the standard LU decomposition that incorporate diagonal and permutation matrices.
This document discusses exponential functions and their properties. It explores exponential growth and decay through graphs of functions like y=2^x and y=0.5^x. It shows that as x increases, exponential growth functions approach infinity, while decay functions approach zero. The document also introduces the irrational number e as the most important base for modeling continuous growth and decay. It shows how the function A=Ce^rt models continuous compound interest as the compounding period approaches infinity.
Series solution of ordinary differential equation
advance engineering mathematics
The power series method is the standard method for solving linear ODEs with variable
coefficients. It gives solutions in the form of power series. These series can be used for computing values, graphing curves, proving formulas, and exploring properties of
solutions, as we shall see.
Antwoorden fourier and laplace transforms, manual solutionsHildemar Alvarez
The document provides solutions to selected exercises from chapter 1 of a textbook. It first shows the steps to express the sum of two time-harmonic signals as another time-harmonic signal. It then defines several integrals involving trigonometric and exponential functions, and calculates their values. Finally, it summarizes properties of linear time-invariant systems and provides example responses to different input signals.
This document appears to be part of a Greek mathematics textbook. It contains definitions of common mathematical terms like function, graphical representation of a function, equality of functions, operations on functions, and composition of functions. It also defines what it means for a function to be increasing or decreasing over an interval of its domain. The document is divided into numbered sections and contains examples to illustrate each definition.
Higher order derivatives are obtained by repeatedly taking the derivative of a function or its derivatives. The order of a derivative refers to how many times differentiation has been performed. To find a higher order derivative, one simply takes the derivative of the existing derivative. For example, to get the third derivative f'''(x) of a function f(x), one would take the derivative of the second derivative f''(x). Higher derivatives provide important information about the curvature and flexibility of a function at different points.
This document discusses OpenStack Neutron and software defined networking. It provides an overview of Neutron and how it allows network as a service capabilities. It describes the packet flow for virtual machines accessing the external network or communicating between virtual machines on the same network. It explains how Neutron integrates with Open vSwitch on the compute nodes to provide networking and discusses the various Neutron agents.
The point of inflection is the point on a curve where the concavity changes, such that the curve transitions from concave down to concave up or vice versa. Mathematically, it is the point where the second derivative equals zero or changes sign, indicating a switch between increasing and decreasing slope. To find the point of inflection, take the second derivative of a function and set it equal to zero to solve for the x-value where the concavity changes.
Σε αυτές τις διαφάνειες ολοκληρώνονται τα μαθήματα σχετικά με το εισαγωγικό κεφάλαιο των συναρτήσεων με την έννοια της αντίστροφης μίας «1-1» συνάρτησης.
The document summarizes existing research on establishing the existence and uniqueness of coupled fixed points for contraction mappings on partially ordered metric spaces. It presents several key theorems:
1) Theorems by Geraghty, Amini-Harandi and Emami, and Gnana Bhaskar and Lakshmikantham establish the existence of unique fixed points for contraction mappings on complete metric spaces and partially ordered metric spaces.
2) Choudhury and Kundu extended these results to Geraghty contractions by introducing an altering distance function.
3) GVR Babu and P. Subhashini further generalized the results to coupled fixed points for Geraghty contractions using an altering distance
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
- The document presents a probabilistic algorithm for computing the polynomial greatest common divisor (PGCD) with smaller factors.
- It summarizes previous work on the subresultant algorithm for computing PGCD and discusses its limitations, such as not always correctly determining the variant τ.
- The new algorithm aims to determine τ correctly in most cases when given two polynomials f(x) and g(x). It does so by adding a few steps instead of directly computing the polynomial t(x) in the relation s(x)f(x) + t(x)g(x) = r(x).
This document provides an overview of relations and their properties in discrete mathematics. It defines what a relation is, distinguishes between relations and functions, and describes key properties of relations including:
- Reflexive relations, where every element is related to itself.
- Symmetric relations, where if a is related to b then b is related to a.
- Transitive relations, where if a is related to b and b is related to c, then a is related to c.
It also discusses how to determine if a relation has these properties, combines multiple relations using set operations, and defines the composite of two relations. The overall goal is for students to understand relations and be able to analyze them for
This document discusses inner product spaces and how inner products can be defined on vector spaces to generalize concepts like the dot product, vector norms, angles between vectors, and distances between vectors. It provides examples of defining inner products on spaces like Rn, the space of polynomials Pn, and the space of 2x2 matrices M22. It shows how norms, orthogonality, and distances can be calculated in these spaces based on their defined inner products. The document also discusses how different inner products can lead to different geometries beyond standard Euclidean geometry.
This document discusses LU decomposition, which is a matrix factorization technique used to solve systems of linear equations. It breaks a matrix down into the product of a lower triangular matrix and an upper triangular matrix. This allows the system of equations to be solved through back substitution and forward substitution. The document also covers LDU, LUP, and LU decompositions with full pivoting, which are variations of the standard LU decomposition that incorporate diagonal and permutation matrices.
This document discusses exponential functions and their properties. It explores exponential growth and decay through graphs of functions like y=2^x and y=0.5^x. It shows that as x increases, exponential growth functions approach infinity, while decay functions approach zero. The document also introduces the irrational number e as the most important base for modeling continuous growth and decay. It shows how the function A=Ce^rt models continuous compound interest as the compounding period approaches infinity.
Series solution of ordinary differential equation
advance engineering mathematics
The power series method is the standard method for solving linear ODEs with variable
coefficients. It gives solutions in the form of power series. These series can be used for computing values, graphing curves, proving formulas, and exploring properties of
solutions, as we shall see.
Antwoorden fourier and laplace transforms, manual solutionsHildemar Alvarez
The document provides solutions to selected exercises from chapter 1 of a textbook. It first shows the steps to express the sum of two time-harmonic signals as another time-harmonic signal. It then defines several integrals involving trigonometric and exponential functions, and calculates their values. Finally, it summarizes properties of linear time-invariant systems and provides example responses to different input signals.
This document appears to be part of a Greek mathematics textbook. It contains definitions of common mathematical terms like function, graphical representation of a function, equality of functions, operations on functions, and composition of functions. It also defines what it means for a function to be increasing or decreasing over an interval of its domain. The document is divided into numbered sections and contains examples to illustrate each definition.
Higher order derivatives are obtained by repeatedly taking the derivative of a function or its derivatives. The order of a derivative refers to how many times differentiation has been performed. To find a higher order derivative, one simply takes the derivative of the existing derivative. For example, to get the third derivative f'''(x) of a function f(x), one would take the derivative of the second derivative f''(x). Higher derivatives provide important information about the curvature and flexibility of a function at different points.
This document discusses OpenStack Neutron and software defined networking. It provides an overview of Neutron and how it allows network as a service capabilities. It describes the packet flow for virtual machines accessing the external network or communicating between virtual machines on the same network. It explains how Neutron integrates with Open vSwitch on the compute nodes to provide networking and discusses the various Neutron agents.
The point of inflection is the point on a curve where the concavity changes, such that the curve transitions from concave down to concave up or vice versa. Mathematically, it is the point where the second derivative equals zero or changes sign, indicating a switch between increasing and decreasing slope. To find the point of inflection, take the second derivative of a function and set it equal to zero to solve for the x-value where the concavity changes.
Σε αυτές τις διαφάνειες ολοκληρώνονται τα μαθήματα σχετικά με το εισαγωγικό κεφάλαιο των συναρτήσεων με την έννοια της αντίστροφης μίας «1-1» συνάρτησης.
The document summarizes existing research on establishing the existence and uniqueness of coupled fixed points for contraction mappings on partially ordered metric spaces. It presents several key theorems:
1) Theorems by Geraghty, Amini-Harandi and Emami, and Gnana Bhaskar and Lakshmikantham establish the existence of unique fixed points for contraction mappings on complete metric spaces and partially ordered metric spaces.
2) Choudhury and Kundu extended these results to Geraghty contractions by introducing an altering distance function.
3) GVR Babu and P. Subhashini further generalized the results to coupled fixed points for Geraghty contractions using an altering distance
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
- The document presents a probabilistic algorithm for computing the polynomial greatest common divisor (PGCD) with smaller factors.
- It summarizes previous work on the subresultant algorithm for computing PGCD and discusses its limitations, such as not always correctly determining the variant τ.
- The new algorithm aims to determine τ correctly in most cases when given two polynomials f(x) and g(x). It does so by adding a few steps instead of directly computing the polynomial t(x) in the relation s(x)f(x) + t(x)g(x) = r(x).
Uniformity of the Local Convergence of Chord Method for Generalized EquationsIOSR Journals
This document summarizes research on the uniform convergence of the Chord method for solving generalized equations. The Chord method is an iterative method for finding solutions to equations of the form y ∈ f(x) + F(x), where f is a function and F is a set-valued mapping. The authors prove that under certain conditions, including F being pseudo-Lipschitz and the derivative of f being continuous, the Chord method converges uniformly for small variations in the parameter y. They obtain this result in two different ways. The document also provides relevant definitions and preliminaries on generalized equations, set-valued mappings, and convergence properties.
Devaney Chaos Induced by Turbulent and Erratic FunctionsIOSRJM
Let I be a compact interval and f be a continuous function defined from I to I. We study the relationship between tubulent function, erratic function and Devaney Chaos.
The document discusses the Gram-Schmidt orthogonalization (GSO) process for constructing an orthonormal basis from a set of linearly independent vectors. It explains that GSO works by taking a vector, normalizing it to unit length to create the first basis vector, then subtracting the component of the next vector along this first vector to make it orthogonal, and repeating this process to iteratively construct an orthonormal basis. An example applies GSO to three vectors in R3, finding the orthonormal basis vectors by removing components along each preceding vector at each step.
Fixed Point Results for Weakly Compatible Mappings in Convex G-Metric Spaceinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
A PROBABILISTIC ALGORITHM OF COMPUTING THE POLYNOMIAL GREATEST COMMON DIVISOR...ijscmcj
In the earlier work, subresultant algorithm was proposed to decrease the coefficient growth in the Euclidean algorithm of polynomials. However, the output polynomial remainders may have a small factor which can be removed to satisfy our needs. Then later, an improved subresultant algorithm was given by representing the subresultant algorithm in another way, where we add a variant called 𝜏 to express the small factor. There was a way to compute the variant proposed by Brown, who worked at IBM. Nevertheless, the way failed to determine each𝜏 correctly.
The document discusses Lie algebras, which are vector spaces with a non-associative multiplication called the Lie bracket. Any Lie group gives rise to a Lie algebra, and vice versa. Lie algebras allow the study of Lie groups in terms of vector spaces. A Lie subalgebra is a vector subspace of a Lie algebra that is closed under the Lie bracket, while an ideal is a subspace where the Lie bracket of any element of the Lie algebra with an element of the ideal is also in the ideal. Examples of Lie algebras and their substructures are provided.
In this paper we consider the initial-boundary value problem for a nonlinear equation induced with respect to the mathematical models in mass production process with the one sided spring boundary condition by boundary feedback control. We establish the asymptotic behavior of solutions to this problem in time, and give an example and simulation to illustrate our results. Results of this paper are able to apply industrial parts such as a typical model widely used to represent threads, wires, magnetic tapes, belts, band saws, and so on.
1) The document presents a wavelet collocation method for numerically solving nth order Volterra integro-differential equations. It expands the unknown function as a series of Chebyshev wavelets of the second kind with unknown coefficients.
2) It states and proves a uniform convergence theorem that establishes the convergence of approximating the solution using truncated Chebyshev wavelet series expansions.
3) The paper demonstrates the validity and applicability of the proposed method through some illustrative examples of solving integro-differential equations using the Chebyshev wavelet collocation approach.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
The document describes the (G'/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations (PDEs) arising in mathematical physics. The method involves expressing the solution as a polynomial in (G'/G), where G satisfies a second order linear ordinary differential equation. The method is demonstrated by using it to find traveling wave solutions of the variable coefficients KdV (vcKdV) equation, the modified dispersive water wave (MDWW) equations, and the symmetrically coupled KdV equations. These solutions are expressed in terms of hyperbolic, trigonometric, and rational functions. The method provides a simple way to obtain exact solutions to important nonlinear PDEs.
This research article discusses using fuzzy logic to enable approximate geometric reasoning with extended objects in incidence geometry. It presents a mathematical framework that allows fuzzification of the axioms and predicates of incidence geometry to account for extended, loosely defined geographic objects. The paper proposes using a specific fuzzy logic system and defines fuzzy predicates for geometric primitives like points and lines. It also introduces fuzzy relations for equivalence of extended lines and uses fuzzy conditional inference to approximate these relations. This framework aims to add capabilities for reasoning with imprecise, extended objects to geographic information systems.
Existence, Uniqueness and Stability Solution of Differential Equations with B...iosrjce
In this work, we investigate the existence ,uniqueness and stability solution of non-linear
differential equations with boundary conditions by using both method Picard approximation and
Banach fixed point theorem which were introduced by [6] .These investigations lead us to improving
and extending the above method. Also we expand the results obtained by [1] to change the non-linear
differential equations with initial condition to non-linear differential equations with boundary
conditions
Second Order Parallel Tensors and Ricci Solitons in S-space forminventionjournals
In this paper, we prove that a symmetric parallel second order covariant tensor in (2m+s)- dimensional S-space form is a constant multiple of the associated metric tensor. Then we apply this result to study Ricci solitons for S-space form and Sasakian space form of dimension 3
Some Common Fixed Point Results for Expansive Mappings in a Cone Metric SpaceIOSR Journals
The purpose of this work is to extend and generalize some common fixed point theorems for Expansive type mappings in complete cone metric spaces. We are attempting to generalize the several well- known recent results. Mathematical subject classification; 54H25, 47H10
A new kind of quantum gates, higher braiding gates, as matrix solutions of the polyadic braid equations (different from the generalized Yang–Baxter equations) is introduced. Such gates lead to another special multiqubit entanglement that can speed up key distribution and accelerate algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates, which can be related with qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, star and circle ones, such that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the introduced classes is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by the higher braid operators are given. Finally, we show, that for each multiqubit state, there exist higher braiding gates that are not entangling, and the concrete conditions to be non-entangling are given for the obtained binary and ternary gates.
Matrix Transformations on Paranormed Sequence Spaces Related To De La Vallée-...inventionjournals
In this paper, we determine the necessary and sufficient conditions to characterize the matrices which transform paranormed sequence spaces into the spaces 푉휎 (휆) and 푉휎 ∞(휆) , where 푉휎 (휆) denotes the space of all (휎, 휆)-convergent sequences and 푉휎 ∞(휆) denotes the space of all (휎, 휆)-bounded sequences defined using the concept of de la Vallée-Pousin mean.
Fixed point theorem between cone metric space and quasi-cone metric spacenooriasukmaningtyas
This study involves new notions of continuity of mapping between quasi-cone metrics spaces (QCMSs), cone metric spaces (CMSs), and vice versa. The relation between all notions of continuity were thoroughly studied and supported with the help of examples. In addition, these new continuities were compared with various types of continuities of mapping between two QCMSs. The continuity types are 𝒇𝒇-continuous, 𝒃𝒃-continuous, 𝒇𝒃-continuous, and 𝒃𝒇-continuous. The results demonstrated that the new notions of continuity could be generalized to the continuity of mapping between two QCMSs. It also showed a fixed point for this continuity map between a complete Hausdorff CMS and QCMS. Overall, this study supports recent research results.
In this paper we introduce the notions of Fuzzy Ideals in BH-algebras and the notion
of fuzzy dot Ideals of BH-algebras and investigate some of their results
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Women with polycystic ovary syndrome (PCOS) have elevated levels of hormones like luteinizing hormone and testosterone, as well as higher levels of insulin and insulin resistance compared to healthy women. They also have increased levels of inflammatory markers like C-reactive protein, interleukin-6, and leptin. This study found these abnormalities in the hormones and inflammatory cytokines of women with PCOS ages 23-40, indicating that hormone imbalances associated with insulin resistance and elevated inflammatory markers may worsen infertility in women with PCOS.
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A Study of Derivations on Lattices
Funmilola Balogun*
Department of Mathematical Sciences and Information Technology, Federal University Dutsin-ma,
Katsina State, Nigeria
* fbalogun@fudutsinma.edu.ng
Abstract
In this paper we introduce the notion 𝑓(𝑥⋀𝑦) = 𝑥⋀𝑓𝑦 , where f is a derivation on a lattice L and 𝑥, 𝑦 ∈ 𝐿,
using this notion we establish equivalence relations on L. Secondly we extend some results of isotone derivations
on a distributive lattice. Finally, it is shown that; 〈𝐹(𝐿),∨,∧〉 is a modular lattice, where L is a modular lattice
and 𝐹(𝐿) is the set of all isotone derivations on L
Keywords: isotone derivation, distributive lattice, modular lattice.
1 Introduction
Several authors [2,3,6,10,11,13,20,22] have studied analytical and algebraic properties of lattice. In the
eighteenth century, George Boole [7] initiated the study of lattices. In this context Richard Dedekind in a series
of paper around 1900, laid foundation for lattice theory. The distributive lattices were introduced by Gratzer [12].
These lattices have provided the motivation for many results in general lattice theory.
Lattice theory has quite a number of applications in many research areas such as information retrieval, and
information access controls(see [8] and[20]). Sandhu [20] showed that lattice based mandatory access controls
can be enforced by appropriate configuration of Role Based Access Control(RBAC) components. In [11], the
author solved several problems in cryptanalysis using tools from the geometry of numbers. The probability
density under a general hypergraphical model was expressed using the co-information lattice in [3].
Derivations in rings and near rings have been widely researched [4,5,15,18,19]. The concept of derivation in a
lattice was introduced in [22], Xin et al characterized modular lattices and distributive lattices by isotone
derivations and gave conditions under which a derivation is isotone for bounded lattices, modular lattices and
distributive lattices. Several other authors like; [1,9,21,23] also studied derivations on lattices. In [21], the author
using fixed sets of isotone derivations established characterizations of a chain, a distributive lattice, a modular
lattice and a relatively pseudo-complimented lattice. The application of the notion of derivation in ring and
near-ring theory to BCI-algebras was given by Jun and Xin [16], see also [24]
The rest of the paper is organized as follows. Section 2, is devoted to some basic definitions and results. In
Section 3, we define 𝑓(𝑥⋀𝑦) = 𝑥⋀𝑓𝑦 and establish equivalence relations using isotone derivations on L.
Section 4 gives an extension of isotone derivations on distributive lattices, which was studied in [22]. Also by
defining a partial order on the set of all isotone derivations on a modular lattice, we prove that this set of isotone
derivations together with the operations of meet ′ ∧ ′ and join ′ ∨ ′ form a modular lattice.
2. Preliminaries
We need some basic definitions and results on lattices and derivations on lattices.
Definition 2.1[6]: Let L be a nonempty set endowed with operations ∧ and ∨. Then (𝐿,∧,∨) is called a lattice
if it satisfies the following conditions for all 𝑥, 𝑦, 𝑧 ∈ 𝐿 :
i. 𝑥 ∧ 𝑥 = 𝑥, 𝑥 ∨ 𝑥 = 𝑥
ii. 𝑥 ∧ 𝑦 = 𝑦 ∧ 𝑥, 𝑥 ∨ 𝑦 = 𝑦 ∨ 𝑥
iii. (𝑥 ∧ 𝑦) ∧ 𝑧 = 𝑥 ∧ (𝑦 ∧ 𝑧), (𝑥 ∨ 𝑦) ∨ 𝑧 = 𝑥 ∨ (𝑦 ∨ 𝑧)
iv. (𝑥 ∧ 𝑦) ∨ 𝑥 = 𝑥, (𝑥 ∨ 𝑦) ∧ 𝑥 = 𝑥.
Definition 2.2 [6]: A lattice (𝐿,∧,∨) is called a distributive lattice if it satisfies any of the following conditions
for all 𝑥, 𝑦, 𝑧 ∈ 𝐿:
v. 𝑥 ∧ (𝑦 ∨ 𝑧) = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)
vi. 𝑥 ∨ (𝑦 ∧ 𝑧) = (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧)
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In any lattice, conditions v. and vi. are equivalent.
Definition 2.3 [2]: A lattice (𝐿,∧,∨) is called a modular lattice if it satisfies the following conditions for all
𝑥, 𝑦, 𝑧 ∈ 𝐿:
vii. If 𝑥 ≤ 𝑧, then 𝑥 ∨ (𝑦 ∧ 𝑧) = (𝑥 ∨ 𝑦) ∧ 𝑧
Condition vii. is called the modular identity.
Example 2.4: A distributive lattice of fundamental importance is the two-element chain(2,∧,∨). This lattice
features prominently in logic as the lattice of truth values.
In [14] the lattice L is called n-distributive if, 𝑥 ∧ (∨𝑖=0
𝑛
𝑦𝑖) =∨𝑖=0
𝑛
(𝑥 ∧ (∨𝑗(≠𝑖)=0
𝑛
𝑦𝑗))
Figure 1 represents a modular lattice.
M5
Fig. 1
Definition 2.5 [6]: Let (𝐿,∧,∨) be a lattice. A binary relation ‘ ≤ ’is defined by 𝑥 ≤ 𝑦 if and only if
𝑥 ∧ 𝑦 = 𝑥 and 𝑥 ∨ 𝑦 = 𝑦.
We need the following result;
Lemma 2.6 [6]: Let (𝐿,∧,∨) be a lattice. Define the binary relation ‘ ≤ ’ as in definition 2.5. Then (𝐿, ≤) is
a partially ordered set (poset) and for any 𝑥, 𝑦 ∈ 𝐿, 𝑥 ∧ 𝑦 is the g.l.b of {𝑥, 𝑦}, and 𝑥 ∨ 𝑦 is the l.u.b of {𝑥, 𝑦}.
From Lemma 2.6, clearly a lattice is an ordered structure.
Definition 2.7 [6]: An ideal I of the lattice (𝐿,∧,∨) is a nonempty subset I of L with the properties:
viii. 𝑥 ≤ 𝑦, 𝑦 ∈ 𝐼 ⟹ 𝑥 ∈ 𝐼
ix. 𝑥, 𝑦 ∈ 𝐼 ⟹ 𝑥 ∨ 𝑦 ∈ 𝐼
𝐼1, 𝐼2 are ideals of L, implies that 𝐼1 ∧ 𝐼2 is an ideal of L.
Definition 2.8 [6]: Let 𝜃: 𝐿 ⟶ 𝑀 be a function from a lattice L to a lattice M. Then 𝜃 is a lattice
homomorphism if 𝜃(𝑥 ∧ 𝑦) = 𝜃(𝑥) ∧ 𝜃(𝑦)
and
𝜃(𝑥 ∨ 𝑦) = 𝜃(𝑥) ∨ 𝜃(𝑦) for all 𝑥, 𝑦 ∈ 𝐿
A homomorphism is called an isomorphism if it is bijective, an epimorphism if it is onto and a monomorphism if
it is one-to-one.
The following is an analogous form of the Leibniz’s formula for derivations in a ring.
Definition 2.9 [22]: Let L be a lattice and 𝑓: 𝐿 → 𝐿 be a function. We call f a derivation on L if it satisfies the
condition:
𝑓(𝑥 ∧ 𝑦) = (𝑓(𝑥) ∧ 𝑦) ∨ (𝑥 ∧ 𝑓(𝑦))
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The following example shows that not all functions are derivations;
Example 2.10 [21]: Take the lattice L in figure 2,
Fig. 2
Define two functions 𝑓1 𝑎𝑛𝑑 𝑓2 on L by
𝑓1 𝛼 = {
𝛼, 𝛼 = 0 𝑜𝑟 1,
𝑦, 𝛼 = 𝑥,
𝑥, 𝛼 = 𝑦
𝑓2 𝛼 = {
𝑥, 𝛼 = 𝑥 𝑜𝑟 1,
0, 𝛼 = 𝑦
0, 𝛼 = 0
𝑓1 is not a derivation but 𝑓2 is a derivation on L.
We have the following result due to [22]
Proposition 2.11: Let L be a lattice and f be a derivation on L. Then the following hold:
1. 𝑓𝑥 ≤ 𝑥;
2. 𝑓𝑥 ∧ 𝑓𝑦 ≤ 𝑓(𝑥 ∧ 𝑦) ≤ 𝑓𝑥 ∨ 𝑓𝑦;
3. If I is an ideal of L, then 𝑓𝐼 ⊆ 𝐼, where 𝑓𝐼 = {𝑓𝑥: 𝑥 ∈ 𝐼};
4. If L has a least element 0, 𝑑0 = 0.
From the result above, it is obvious that derivations in lattices are contraction mappings i.e. 𝑓𝑥 ≤ 𝑥.
Next we have the following results;
Definition 2.12 [22]: Let L be a lattice and f be a derivation on L. If 𝑥 ≤ 𝑦 implies 𝑓𝑥 ≤ 𝑓𝑦, then f is called an
isotone derivation.
Proposition 2.13 [22]: Let L be a lattice and f be a derivation on L. If 𝑦 ≤ 𝑥 and 𝑓𝑥 = 𝑥, then 𝑓𝑦 = 𝑦.
Theorem 2.14 [22]: Let L be a distributive lattice and f be a derivation on L. The following conditions are
equivalent:
1. f is an isotone derivation;
2. 𝑓(𝑥 ∧ 𝑦) = 𝑓𝑥 ∧ 𝑓𝑦;
3. 𝑓(𝑥 ∨ 𝑦) = 𝑓𝑥 ∨ 𝑓𝑦
3. Derivations on Lattices
The following result is as presented in [21]: Let L be a lattice and 𝑎 ∈ 𝐿. Define a function 𝑓𝑎 by 𝑓𝑎(𝑥) = 𝑥 ∧
𝑎 for all 𝑥 ∈ 𝐿 then 𝑓𝑎 is a derivation on L. Such derivations are called principal derivations.
We include the proof of the above for the sake of convenience for the reader.
𝑓𝑎(𝑥 ∧ 𝑦) = (𝑓𝑎 𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑓𝑎 𝑦) = ((𝑥 ∧ 𝑎) ∧ 𝑦) ∨ (𝑥 ∧ (𝑦 ∧ 𝑎)) = ((𝑥 ∧ 𝑦) ∧ 𝑎) ∨ ((𝑥 ∧ 𝑦) ∧ 𝑎) =
(𝑥 ∧ 𝑦) ∧ (𝑎 ∨ 𝑎) = (𝑥 ∧ 𝑦) ∧ 𝑎 □
Motivated by the observation above, we introduce the following;
Proposition 3.2: Let L be a lattice and 𝑎 ∈ 𝐿. Define a function 𝑓𝑎 by 𝑓𝑎(𝑥) = 𝑥 ∨ 𝑎 for all 𝑥 ∈ 𝐿, then
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𝑓𝑎 is a derivation (also a principal derivation) on L.
𝑓𝑎(𝑥 ∨ 𝑦) = 𝑓𝑎 𝑥 ∨ 𝑓𝑎 𝑦 = (𝑥 ∨ 𝑎) ∨ (𝑦 ∨ 𝑎) = 𝑥 ∨ (𝑎 ∨ (𝑦 ∨ 𝑎)) = 𝑥 ∨ ((𝑎 ∨ 𝑦) ∨ 𝑎) = 𝑥 ∨ (𝑦 ∨ 𝑎) ∨ 𝑎
= (𝑥 ∨ 𝑦) ∨ (𝑎 ∨ 𝑎) = (𝑥 ∨ 𝑦) ∨ 𝑎 □
Proposition 3.3: Every principal derivation of a lattice L is an isotone derivation of L.
Proof
Let 𝑓𝑎 be a principal derivation of a lattice L. Since for any 𝑥, 𝑦 ∈ 𝐿, we have 𝑓𝑎 𝑥 = 𝑥 ∧ 𝑎 ≤ 𝑦 ∧ 𝑎 = 𝑓𝑎 𝑦 and
hence 𝑓𝑎 is isotone.(see[21] for details)
Similarly, if we define 𝑓𝑎 = 𝑥 ∨ 𝑎, we have;
For any 𝑥, 𝑦 ∈ 𝐿 𝑎𝑛𝑑 𝑥 ≤ 𝑦, 𝑓𝑎 𝑥 = 𝑥 ∨ 𝑎 ≤ 𝑦 ∨ 𝑎 = 𝑓𝑎 𝑦 and hence 𝑓𝑎 is isotone. □
Next we prove the following results;
Theorem 3.4: Let L be a lattice and 𝑓: 𝐿 → 𝐿 be a derivation. The following conditions are equivalent:
1. f is an isotone derivation
2. 𝑓(𝑥 ∧ 𝑦) = 𝑥 ∧ 𝑓𝑦
Proof
1 ⟹ 2
Suppose f is an isotone derivation,
we have; 𝑥 ≤ 𝑦 ⟹ 𝑓𝑥 ≤ 𝑓𝑦, ∀𝑥, 𝑦 ∈ 𝐿
Then 𝑓(𝑥 ∧ 𝑦) = (𝑓𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑓𝑦) ≥ 𝑥 ∧ 𝑓𝑦 … … … (𝑖)
Also, 𝑥 ∧ 𝑦 ≤ 𝑥 and 𝑥 ∧ 𝑦 ≤ 𝑦
This implies that, 𝑓(𝑥 ∧ 𝑦) ≤ 𝑓𝑥 and 𝑓(𝑥 ∧ 𝑦) ≤ 𝑓𝑦
We have, 𝑓(𝑥 ∧ 𝑦) ≤ 𝑓𝑥 ∧ 𝑓𝑦 ≤ 𝑥 ∧ 𝑓𝑦 … … … … … … … (𝑖𝑖)
From (i) and (ii) we have 𝑓(𝑥 ∧ 𝑦) = 𝑥 ∧ 𝑓𝑦
2 ⟹ 1
Suppose 𝑓(𝑥 ∧ 𝑦) = 𝑥 ∧ 𝑓𝑦 ∀𝑥, 𝑦 ∈ 𝐿
We have; 𝑓(𝑥 ∧ 𝑦) = (𝑥 ∧ 𝑓𝑦) ∨ (𝑓𝑥 ∧ 𝑦) = 𝑥 ∧ 𝑓𝑦
If 𝑥 ≤ 𝑦, 𝑠𝑖𝑛𝑐𝑒 𝑓(𝑥 ∧ 𝑦) = 𝑥 ∧ 𝑓𝑦 ⟹ 𝑓𝑥 = 𝑥 ∧ 𝑓𝑦
We have 𝑓𝑥 ∨ 𝑓𝑦 = (𝑥 ∧ 𝑓𝑦) ∨ 𝑓𝑦 = 𝑓𝑦
Therefore, 𝑓𝑥 ≤ 𝑓𝑦, hence f is isotone. □
Theorem 3.5: Let L be a lattice and 𝑓: 𝐿 ⟶ 𝐿 be a derivation. The following conditions are equivalent:
1. 𝑓(𝑥 ∧ 𝑦) = 𝑥 ∧ 𝑓𝑦;
2. 𝑓(𝑥 ∧ 𝑦) = 𝑓𝑥 ∧ 𝑓𝑦
Proof
(1)⟹ (2)
Suppose 𝑓(𝑥 ∧ 𝑦) = 𝑥 ∧ 𝑓𝑦, clearly, 𝑓𝑥 ∧ 𝑓𝑦 ≤ 𝑓𝑥 ∧ 𝑦
Since 𝑥 ∧ 𝑓𝑦 = 𝑓(𝑥 ∧ 𝑦) also, 𝑓𝑦 ∧ 𝑥 ≤ 𝑓𝑦 and 𝑓𝑥 ∧ 𝑦 ≤ 𝑓𝑥
This implies that 𝑓𝑦 ∧ 𝑥 = 𝑓𝑥 ∧ 𝑦 ≤ 𝑓𝑥 ∧ 𝑓𝑦 ⟹ 𝑥 ∧ 𝑓𝑦 = 𝑦 ∧ 𝑓𝑥 ≤ 𝑓𝑥 ∧ 𝑓𝑦
Therefore, 𝑓(𝑥 ∧ 𝑦) = 𝑓𝑥 ∧ 𝑓𝑦.
(2)⟹ (1)
Suppose 𝑓(𝑥 ∧ 𝑦) = 𝑓𝑥 ∧ 𝑓𝑦 ∀𝑥, 𝑦 ∈ 𝐿
If 𝑥 ≤ 𝑦 then 𝑓𝑥 = 𝑓(𝑥 ∧ 𝑦) = 𝑓𝑥 ∧ 𝑓𝑦 ⟹ 𝑓𝑥 ≤ 𝑓𝑦, hence f is an isotone derivation. Since f is an isotone
derivation, by Theorem 3.4, we have,𝑓(𝑥 ∧ 𝑦) = 𝑥 ∧ 𝑓𝑦 □
From Theorems 3.4 and 3.5 we have the following result;
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Theorem 3.6: Let L be a lattice and 𝑓: 𝐿 → 𝐿 be a derivation. The following conditions are equivalent;
1. f is an isotone derivation;
2. 𝑓(𝑥 ∧ 𝑦) = 𝑥 ∧ 𝑓𝑦
3. 𝑓(𝑥 ∧ 𝑦) = 𝑓𝑥 ∧ 𝑓𝑦
4. Derivations on Distributive and Modular Lattices
Distributive lattices have provided the motivation for many results in general lattice theory. Many conditions on
lattices are weakend forms of distributivity. Hence derivations on distributive lattices have stronger properties.
Theorem 4.1 [21]: Let L be a distributive lattice and 𝑓1 𝑎𝑛𝑑 𝑓2 be two isotone derivations on L.
Define
(𝑓1 ∧ 𝑓2)(𝑥) = 𝑓1 𝑥 ∧ 𝑓2 𝑥,
(𝑓1 ∨ 𝑓2)(𝑥) = 𝑓1 𝑥 ∨ 𝑓2 𝑥.
Then 𝑓1 ∧ 𝑓2 and 𝑓1 ∨ 𝑓2 are also isotone derivation on L.
In this sequel we establish the following result;
Theorem 4.2: Let L be a distributive lattice. Let 𝑓1, 𝑓2 and 𝑓3 be isotone derivations on L, defined by
((𝑓1 ∧ 𝑓2) ∧ 𝑓3)𝑥 = (𝑓1 𝑥 ∧ 𝑓2 𝑥) ∧ 𝑓3 𝑥
((𝑓1 ∨ 𝑓2) ∨ 𝑓3)𝑥 = (𝑓1 𝑥 ∨ 𝑓2 𝑥) ∨ 𝑓3 𝑥
Then (𝑓1 ∧ 𝑓2) ∧ 𝑓3 and (𝑓1 ∨ 𝑓2) ∨ 𝑓3 are also isotone derivations on L.
Proof
We begin with,
((𝑓1 ∨ 𝑓2) ∨ 𝑓3)(𝑥 ∧ 𝑦) = 𝑓1(𝑥 ∧ 𝑦) ∨ 𝑓2(𝑥 ∧ 𝑦) ∨ 𝑓3(𝑥 ∧ 𝑦)
= (𝑥 ∧ 𝑓1 𝑦) ∨ (𝑥 ∧ 𝑓2 𝑦) ∨ (𝑥 ∧ 𝑓3 𝑦) = 𝑥 ∧ ((𝑓1 𝑦 ∨ 𝑓2 𝑦) ∨ 𝑓3 𝑦)
= 𝑥 ∧ ((𝑓1 ∨ 𝑓2) ∨ 𝑓3)𝑦 … … … … … … … … … … … … … … … … … … … (𝑖)
Similarly we have,
((𝑓1 ∨ 𝑓2) ∨ 𝑓3)(𝑥 ∧ 𝑦) = ((𝑓1 ∨ 𝑓2) ∨ 𝑓3)𝑥 ∧ 𝑦 … … … … … … … … (𝑖𝑖)
Combining (i) and (ii) we get
((𝑓1 ∨ 𝑓2) ∨ 𝑓3)(𝑥 ∧ 𝑦) = (((𝑓1 ∨ 𝑓2) ∨ 𝑓3)𝑥 ∧ 𝑦) ∨ (((𝑓1 ∨ 𝑓2) ∨ 𝑓3)𝑦 ∧ 𝑥)
Therefore (𝑓1 ∨ 𝑓1) ∨ 𝑓1 is a derivation on L.
Also, by Theorem 2.14 (𝑓1 ∨ 𝑓2) ∨ 𝑓3 is isotone since
((𝑓1 ∨ 𝑓2) ∨ 𝑓3)(𝑥 ∨ 𝑦) = ((𝑓1 ∨ 𝑓2)(𝑥 ∨ 𝑦)) ∨ 𝑓3(𝑥 ∨ 𝑦) = 𝑓1(𝑥 ∨ 𝑦) ∨ 𝑓2(𝑥 ∨ 𝑦) ∨ 𝑓3(𝑥 ∨ 𝑦)
= (𝑓1 𝑥 ∨ 𝑓1 𝑦) ∨ (𝑓2 𝑥 ∨ 𝑓2 𝑦) ∨ (𝑓3 𝑥 ∨ 𝑓3 𝑦) = ((𝑓1 𝑥 ∨ 𝑓2 𝑥) ∨ 𝑓3 𝑥) ∨ ((𝑓1 𝑦 ∨ 𝑓2 𝑦) ∨ 𝑓3 𝑦)
= ((𝑓1 ∨ 𝑓2)𝑥 ∨ 𝑓3 𝑥) ∨ ((𝑓1 ∨ 𝑓2)𝑦 ∨ 𝑓3 𝑦) = ((𝑓1 ∨ 𝑓2) ∨ 𝑓3)𝑥 ∨ ((𝑓1 ∨ 𝑓2) ∨ 𝑓3)𝑦
Similarly (𝑓1 ∧ 𝑓2) ∧ 𝑓3 is an isotone derivation on L. □
The following result is due to [21]
Theorem 4.3: Let L be a distributive lattice and 𝐷(𝐿) be a set of all isotone derivations on L. Then
〈𝐷(𝐿),∨,∧〉 is a distributive lattice.
In this sequel we give the following result for modular lattices.
Theorem 4.4: Let L be a modular lattice and 𝐹(𝐿) be a set of all isotone derivations on L.
Then 〈𝐹(𝐿),∨,∧〉 is a modular lattice.
Proof
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From Theorem 4.2, we know that ∧ 𝑎𝑛𝑑 ∨ are binary operators on 𝐹(𝐿). Define a partial order ‘ ≤ ’ on 𝐹(𝐿)
by 𝑓1 ≤ 𝑓2 if and only if 𝑓1 ∧ 𝑓2 = 𝑓1 𝑎𝑛𝑑 𝑓1 ∨ 𝑓2 = 𝑓2.
The g.l.b {𝑓1, 𝑓2} = 𝑓1 ∧ 𝑓2, and l.u.b {𝑓1, 𝑓2} = 𝑓1 ∨ 𝑓2,
hence 〈𝐹(𝐿),∨,∧〉 is a lattice.
Furthermore, for any 𝑓1, 𝑓2, 𝑓3 ∈ 𝐹(𝐿) and for all 𝑥 ∈ 𝐿 we have,
(𝑓1 ∨ (𝑓2 ∧ 𝑓3))𝑥 = 𝑓1 𝑥 ∨ (𝑓2 𝑥 ∧ 𝑓3 𝑥) = (𝑓1 𝑥 ∨ 𝑓2 𝑥) ∧ (𝑓1 𝑥 ∨ 𝑓3 𝑥) = ((𝑓1 ∨ 𝑓2)𝑥) ∧ ((𝑓1 ∨ 𝑓3)𝑥) = 𝑓2 𝑥 ∧ 𝑓3 𝑥
= (𝑓2 ∧ 𝑓3)𝑥 = 𝑓2 𝑥
also,
((𝑓1 ∨ 𝑓2) ∧ 𝑓3)𝑥 = (𝑓1 𝑥 ∨ 𝑓2 𝑥) ∧ 𝑓3 𝑥 = (𝑓1 𝑥 ∧ 𝑓3 𝑥)𝑥 ∨ (𝑓2 𝑥 ∧ 𝑓3 𝑥) = (𝑓1 ∧ 𝑓3)𝑥 ∨ (𝑓2 ∧ 𝑓3)𝑥 = 𝑓1 𝑥 ∨ 𝑓2 𝑥
= (𝑓1 ∨ 𝑓2)𝑥 = 𝑓2 𝑥
Therefore, (𝑓1 ∨ (𝑓2 ∧ 𝑓3))𝑥 = ((𝑓1 ∨ 𝑓2) ∧ 𝑓3)𝑥 □
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