scholarly journals Pelabelan Lokal Titik Graf Hasil Diagram Lattice Subgrup Zn

2018 ◽  
Vol 6 (1) ◽  
pp. 47-56
Author(s):  
Ikhsanul Halikin
Keyword(s):  
Binary Operation ◽  
Identity Element ◽  
Partial Ordering

AbstractA group is a system that contains a set and a binary operation satisfying four axioms, i.e., the set is closed under binary operation, associative, has an identity element, and each element has an inverse. Since the group is essentially a set and the set itself has subsets, so if the binary operation is applied to its subsets then it satisfies the group's four axioms, the subsets with the binary operation are called subgroups. The group and subgroups further form a partial ordering relation. Partial ordering relation is a relation that has reflexive, antisymmetric, and transitive properties. Since the connection of subgroups of a group is partial ordering relation, it can be drawn a lattice diagram. The set of integers modulo n, , is a group under addition modulo n. If the subgroups of are represented as vertex and relations that is connecting two subgroups are represented as edgean , then a graph is obtained. Furthermore, the vertex in this graph can be labeled by their subgroup elements. In this research, we get the result about the characteristic of the lattice diagram of and the existence of vertex local labeling.AbstrakGrup merupakan sistem yang memuat sebuah himpunan dan operasi biner yang memenuhi 4 aksioma, yaitu operasi pada himpunannya bersifat tertutup, assosiatif, memiliki elemen identitas, dan setiap elemennya memiliki invers. Grup pada dasarnya adalah himpunan dan himpunan itu memiliki himpunan bagian. Jika operasi tersebut diberlakukan pada himpunan bagiannya dan memenuhi 4 aksioma grup maka himpunan bagian dan operasi tersebut disebut subgrup. Grup dan subgrup ini selanjutnya membentuk suatu relasi pengurutan parsial. Relasi pengurutan parsial adalah suatu relasi yang memiliki sifat refleksif, antisimetris, dan transitif. Oleh karenanya, relasi subgrup-subgrup dari suatu grup ini dapat digambar diagram latticenya. Himpunan bilangan bulat modulo n, , merupakan grup terhadap operasi penjumlahan modulo n. Jika subgrup pada direpresentasikan sebagai titik dan relasi yang menghubungkan dua buah subgrupnya direpresentasikan sebagai sisi, maka diperoleh suatu graf. Titik-titik pada graf ini dapat dilabeli berdasarkan elemen-elemen subgrupnya. Pada penelitian ini diperoleh hasil kajian mengenai karakteristik diagram lattice subgrup dan eksistensi pelabelan lokal titiknya.

On Anti-Commutative Algebras and Analytic Loops

1965 ◽  
Vol 17 ◽  
pp. 550-558 ◽  
Author(s):  
Arthur A. Sagle
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Binary Operation ◽  
Identity Element ◽  
Unique Element ◽  
Commutative Algebras ◽  
Algebraic Properties

In (4) Malcev generalizes the notion of the Lie algebra of a Lie group to that of an anti-commutative "tangent algebra" of an analytic loop. In this paper we shall discuss these concepts briefly and modify them to the situation where the cancellation laws in the loop are replaced by a unique two-sided inverse. Thus we shall have a set H with a binary operation xy defined on it having the algebraic properties(1.1) H contains a two-sided identity element e;(1.2) for every x ∊ H, there exists a unique element x-1 ∊ H such that xx-1 = x-1x = e;

Injective Hulls of Semilattices

1970 ◽  
Vol 13 (1) ◽  
pp. 115-118 ◽  
Author(s):  
G. Bruns ◽  
H. Lakser
Keyword(s):  
Upper Bound ◽  
Partially Ordered Set ◽  
Binary Operation ◽  
Ordered Set ◽  
Upper Bounds ◽  
Partial Ordering ◽  
Partially Ordered ◽  
Image Position ◽  
Injective Hulls

A (meet-) semilattice is an algebra with one binary operation ∧, which is associative, commutative and idempotent. Throughout this paper we are working in the category of semilattices. All categorical or general algebraic notions are to be understood in this category. In every semilattice S the relationdefines a partial ordering of S. The symbol "∨" denotes least upper bounds under this partial ordering. If it is not clear from the context in which partially ordered set a least upper bound is taken, we add this set as an index to the symbol; for example, ∨AX denotes the least upper bound of X in the partially ordered set A.

GENERALIZED COMPOSITIONS OF FUZZY RELATIONS

2002 ◽  
Vol 10 (supp01) ◽  
pp. 149-163 ◽  
Author(s):  
JÓZEF DREWNIAK ◽  
KRZYSZTOF KULA
Keyword(s):  
Binary Operation ◽  
Identity Element ◽  
Fuzzy Relations ◽  
Triangular Norms ◽  
Algebraic Properties

We examine compositions of fuzzy relations based on a binary operation *. We discuss the dependences between algebraic properties of the operation * and the induced sup –* composition. It is examined independently for monotone operations, for operations with idempotent, zero or identity element, for distributive and associative operations. Finally, we present consequences of these results for compositions based on triangular norms, triangular conorms and uninorms.

An axiomatisation of quantum logic

1973 ◽  
Vol 38 (3) ◽  
pp. 389-392 ◽  
Author(s):  
Ian D. Clark
Keyword(s):  
Quantum Logic ◽  
Orthomodular Lattice ◽  
Binary Operation ◽  
Ordered Set ◽  
Unary Operation ◽  
Axiom System ◽  
Partial Ordering ◽  
Orthomodular Poset ◽  
Implication Algebra ◽  
Image Position

The purpose of this paper is to give an axiom system for quantum logic. Here quantum logic is considered to have the structure of an orthomodular lattice. Some authors assume that it has the structure of an orthomodular poset.In finding this axiom system the implication algebra given in Finch [1] has been very useful. Finch shows there that this algebra can be produced from an orthomodular lattice and vice versa.Definition. An orthocomplementation N on a poset (partially ordered set) whose partial ordering is denoted by ≤ and which has least and greatest elements 0 and 1 is a unary operation satisfying the following:(1) the greatest lower bound of a and Na exists and is 0,(2) a ≤ b implies Nb ≤ Na,(3) NNa = a.Definition. An orthomodular lattice is a lattice with meet ∧, join ∨, least and greatest elements 0 and 1 and an orthocomplementation N satisfyingwhere a ≤ b means a ∧ b = a, as usual.Definition. A Finch implication algebra is a poset with a partial ordering ≤, least and greatest elements 0 and 1 which is orthocomplemented by N. In addition, it has a binary operation → satisfying the following:An orthomodular lattice gives a Finch implication algebra by defining → byA Finch implication algebra can be changed into an orthomodular lattice by defining the meet ∧ and join ∨ byThe orthocomplementation is unchanged in both cases.

Expansions of dense linear orders with the intermediate value property

2001 ◽  
Vol 66 (4) ◽  
pp. 1783-1790 ◽  
Author(s):  
Chris Miller
Keyword(s):  
Binary Operation ◽  
Identity Element ◽  
Linear Orders ◽  
Ordered Group ◽  
Definable Sets ◽  
Noncyclic Subgroup ◽  
Preceding Paragraph ◽  
Intermediate Value ◽  
Familiar Result ◽  
Definable Function

Let ℜ be an expansion of a dense linear order (R, <) without endpoints having the intermediate value property, that is, for all a, b ∈ R, every continuous (parametrically) definable function f: [a, b] → R takes on all values in R between f(a) and f(b). Every expansion of the real line (ℝ, <), as well as every o-minimal expansion of (R, <), has the intermediate value property. Conversely, some nice properties, often associated with expansions of (ℝ, <) or with o-minimal structures, hold for sets and functions definable in ℜ. For example, images of closed bounded definable sets under continuous definable maps are closed and bounded (Proposition 1.10).Of particular interest is the case that ℜ expands an ordered group, that is, ℜ defines a binary operation * such that (R, <, *) is an ordered group. Then (R, *) is abelian and divisible (Proposition 2.2). Continuous nontrivial definable endo-morphisms of (R, *) are surjective and strictly monotone, and monotone nontrivial definable endomorphisms of (R, *) are strictly monotone, continuous and surjective (Proposition 2.4). There is a generalization of the familiar result that every proper noncyclic subgroup of (ℝ, +) is dense and codense in ℝ: If G is a proper nontrivial subgroup of (R, *) definable in ℜ, then either G is dense and codense in R, or G contains an element u such that (R, <, *, e, u, G) is elementarily equivalent to (ℚ, <, +, 0, 1, ℤ), where e denotes the identity element of (R, *) (Theorem 2.3).Here is an outline of this paper. First, we deal with some basic topological results. We then assume that ℜ expands an ordered group and establish the results mentioned in the preceding paragraph. Some examples are then given, followed by a brief discussion of analytic results and possible limitations. In an appendix, an explicit axiomatization (used in the proof of Theorem 2.3) is given for the complete theory of the structure (ℚ, <, +, 0, 1, ℤ).

On endomorphisms of power-semigroups

2017 ◽  
Vol 10 (03) ◽  
pp. 1750058 ◽  
Author(s):  
Yeni Susanti ◽  
Joerg Koppitz
Keyword(s):  
Binary Operation ◽  
Identity Element ◽  
Orthodox Semigroup ◽  
Natural Way

An involuted semilattice [Formula: see text] is a semilattice [Formula: see text] with an identity element [Formula: see text] and with an involution [Formula: see text] satisfying [Formula: see text] and [Formula: see text]. We consider involuted semilattices [Formula: see text] with an identity [Formula: see text] such that there is a subsemilattice [Formula: see text] without [Formula: see text] with the property that any [Formula: see text] belongs to exactly one of the following four sets : [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. In this paper, we introduce an associative binary operation [Formula: see text] on [Formula: see text] in the following quite natural way: [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text] and characterize all endomorphisms of the orthodox semigroup [Formula: see text].

ON QUOTIENT STRUCTURE OF TAKASAKI QUANDLES

2013 ◽  
Vol 22 (12) ◽  
pp. 1341001 ◽  
Author(s):  
YONGJU BAE ◽  
SEONGJEONG KIM
Keyword(s):  
Abelian Group ◽  
Binary Operation ◽  
Identity Element ◽  
Algebraic Properties

A Takasaki quandle is defined by the binary operation a * b = 2b - a on an abelian group G. A Takasaki quandle depends on the algebraic properties of the underlying abelian group. In this paper, we will study the quotient structure of a Takasaki quandle in terms of its subquandle. If a subquandle X of a quandle Q is a subgroup of the underlying group Q, then we can define the quandle structure on the set {X * g | g ∈ Q}, which is called the quotient quandle of Q. Also we will study conditions for a subquandle X to be a subgroup of the underlying group when it contains the identity element.

Loop-theoretic properties of H-spaces

1991 ◽  
Vol 110 (1) ◽  
pp. 121-136 ◽  
Author(s):  
Martin Arkowitz ◽  
Gregory Lupton
Keyword(s):  
Topological Space ◽  
Classical Result ◽  
Binary Operation ◽  
Identity Element ◽  
Base Point ◽  
The Other ◽  
Homotopy Classes ◽  
Algebraic Loop

If X is a topological space with base point, then a based map m: X × X → X is called a multiplication if m restricted to each factor of X × X is homotopic to the identity map of X. The pair (X, m) is then called an H-space. If A is a based topological space and (X, m) is an H-space, then the multiplication m induces a binary operation on the set [A, X] of based homotopy classes of maps of A into X. A classical result due to James [6, theorem 1·1 asserts that if A is a CW-complex and (X, m) is an H-space, then the binary operation gives [A, X] the structure of an algebraic loop. That is, [A, X] has a two-sided identity element and if a, b ∈[A, X], then the equations ax = b and ya = b have unique solutions x, y ∈ [A, X]. Thus it is meaningful to consider loop-theoretic properties of H-spaces. In this paper we make a detailed study of the following loop-theoretic notions applied to H-spaces: inversivity, power-associativity, quasi-commutativity and the Moufang property – see Section 2 for the definitions. If an H-space is Moufang, then it has the other three properties. Moreover, an associative loop is Moufang, and so a homotopy-associative H-space has all four properties. Since many of the standard H-spaces are homotopy-associative, we are particularly interested in determining when an H-space, in particular a finite CW H-space, does not have one of these properties.

Babbitt’s Beguiling Surfaces, Improvised Inside; Part I: Freedoms

2019 ◽  
Vol 5 (1) ◽  
Author(s):  
Joshua Banks Mailman
Keyword(s):  
Partial Ordering ◽  
Personal Traits ◽  
Milton Babbitt ◽  
Iconic Figure ◽  
Inside Part

Milton Babbitt has been a controversial and iconic figure, which has indirectly led to fallacious assumptions about how his music is made, and therefore to fundamental misconceptions about how it might be heard and appreciated. This video (the first of a three-part video essay) reconsiders his music in light of both his personal traits and a more precise examination of the constraints and freedoms entailed by his unusual and often misunderstood compositional practices, which are based inherently on partial ordering (as well as pitch repetition), which enables a surprising amount of freedom to compose the surface details we hear. The opening of Babbitt’s Composition for Four Instruments (1948) and three recompositions (based on re-ordering of pitches) demonstrate the freedoms intrinsic to partial ordering.

Minimal distributions in a stochastic partial ordering and bounds for Gaussian processes

1983 ◽  
Vol 20 (03) ◽  
pp. 529-536
Author(s):  
W. J. R. Eplett
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Partial Ordering ◽  
Boundary Crossing ◽  
Conditional Probabilities ◽  
Life Distribution ◽  
Life Distributions ◽  
Bounded Below ◽  
Crossing Probabilities

A natural requirement to impose upon the life distribution of a component is that after inspection at some randomly chosen time to check whether it is still functioning, its life distribution from the time of checking should be bounded below by some specified distribution which may be defined by external considerations. Furthermore, the life distribution should ideally be minimal in the partial ordering obtained from the conditional probabilities. We prove that these specifications provide an apparently new characterization of the DFRA class of life distributions with a corresponding result for IFRA distributions. These results may be transferred, using Slepian's lemma, to obtain bounds for the boundary crossing probabilities of a stationary Gaussian process.

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