On direct decompositions. I

1952 ◽  
Vol 48 (1) ◽  
pp. 1-22 ◽  
Author(s):  
A. W. Goldie
Keyword(s):  
Group Theory ◽  
Abelian Group ◽  
Binary Operation ◽  
The Other ◽  
Large Classes ◽  
Finite Algebras ◽  
Chain Conditions ◽  
Other Hand ◽  
Decomposition Operator ◽  
Direct Decompositions

It is known that the refinement theorems for direct decompositions, now classical for group theory, are true, under suitable chain conditions, for two large classes of algebras. These are (1) the algebras whose congruences commute and (2) the algebras with a binary operation of a particular type generated by what we shall call a decomposition operator. On the other hand, there exist finite algebras for which the refinement theorems are not true. The problem of characterizing algebras for which the theorems are true arises at once. It is difficult, because the structures of algebras of classes (1) and (2) can be, to a large extent, incompatible. To illustrate this, we mention that any algebra of (2) has a naturally defined centre which is an Abelian group under the binary operation.

On quotient structure of Takasaki quandles II

2014 ◽  
Vol 23 (07) ◽  
pp. 1460012 ◽  
Author(s):  
Yongju Bae ◽  
Seongjeong Kim
Keyword(s):  
Abelian Group ◽  
Knot Theory ◽  
Binary Operation ◽  
The Other ◽  
Other Hand

A Takasaki quandle (T(G), *) is a quandle under the binary operation * defined by a*b = 2b-a for an abelian group (G, +). In this paper, we will show that if a subquandle X of a Takasaki quandle G is a image of subgroup of G under a quandle automorphism of T(G), then the set {X * g | g ∈ G} is a quandle under the binary operation *′ defined by (X * g) *′ (X * h) = X * (g * h). On the other hand, the quotient structure studied in [On quotients of quandles, J. Knot Theory Ramifications 19(9) (2010) 1145–1156] can be applied to the Takasaki quandles. In this paper, we will review the quotient structure studied in [On quotients of quandles, J. Knot Theory Ramifications 19(9) (2010) 1145–1156], and show that the quotient quandle coincides with the quotient quandle defined by Bunch, Lofgren, Rapp and Yetter in [On quotients of quandles, J. Knot Theory Ramifications 19(9) (2010) 1145–1156] for connected Takasaki quandles.

scholarly journals Isometries and direct decompositions of pseudo MV-algebras

2007 ◽  
Vol 57 (2) ◽  
Author(s):  
Milan Jasem
Keyword(s):  
Direct Decomposition ◽  
The Other ◽  
Other Hand ◽  
Mv Algebra ◽  
Direct Decompositions ◽  
Mv Algebras

AbstractIn the paper isometries in pseudo MV-algebras are investigated. It is shown that for every isometry f in a pseudo MV-algebra $$\mathcal{A}$$ = (A, ⊕, −, ∼, 0, 1) there exists an internal direct decomposition $$\mathcal{A} = \mathcal{B}^0 \times \mathcal{C}^0 $$ of $$\mathcal{A}$$ with $$\mathcal{C}^0 $$ commutative such that $$f(0) = 1_{C^0 } $$ and $$f(x) = x_{B^0 } \oplus (1_{C^0 } \odot (x_{C^0 } )^ - ) = x_{B^0 } \oplus (1_{C^0 } - x_{C^0 } )$$ for each x ∈ A.On the other hand, if $$\mathcal{A} = \mathcal{P}^0 \times \mathcal{Q}^0 $$ is an internal direct decomposition of a pseudo MV-algebra $$\mathcal{A}$$ = (A, ⊕, −, ∼, 0, 1) with $$\mathcal{Q}^0 $$ commutative, then the mapping g: A → A defined by $$g(x) = x_{P^0 } \oplus (1_{Q^0 } - x_{Q^0 } )$$ is an isometry in $$\mathcal{A}$$ and $$g(0) = 1_{Q^0 } $$ .

Automorphism groups with some finiteness conditions

2017 ◽  
Vol 67 (4) ◽  
Author(s):  
Konrad Pióro
Keyword(s):  
Lower Bound ◽  
Binary Operation ◽  
Automorphism Groups ◽  
The Other ◽  
Finiteness Conditions ◽  
Other Hand ◽  
The Right

AbstractAll considered groups are torsion or do not contain infinitely generated subgroups. If such a groupNext, we show that ifThe Birkhoff’s construction can be slightly modified so as to obtain a smaller set of operations. In fact, it is enough to take the right multiplications by generators. Moreover, we show that this is the best possible lower bound for the number of unary operations in the case of groups considered here. If we admit non-unary operations, then for finite and countable groups we can reduce the number of operations to one binary operation. On the other hand, if

scholarly journals Zero-sum subsets of decomposable sets in Abelian groups

2020 ◽  
Vol 30 (1) ◽  
pp. 15-25
Author(s):  
T. Banakh ◽  
◽  
A. Ravsky ◽  
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Open Problem ◽  
Abelian Groups ◽  
The Other ◽  
Other Hand ◽  
Decomposable Sets ◽  
Empty Subset ◽  
Zero Sum

A subset D of an abelian group is decomposable if ∅≠D⊂D+D. In the paper we give partial answers to an open problem asking whether every finite decomposable subset D of an abelian group contains a non-empty subset Z⊂D with ∑Z=0. For every n∈N we present a decomposable subset D of cardinality |D|=n in the cyclic group of order 2n−1 such that ∑D=0, but ∑T≠0 for any proper non-empty subset T⊂D. On the other hand, we prove that every decomposable subset D⊂R of cardinality |D|≤7 contains a non-empty subset T⊂D of cardinality |Z|≤12|D| with ∑Z=0. For every n∈N we present a subset D⊂Z of cardinality |D|=2n such that ∑Z=0 for some subset Z⊂D of cardinality |Z|=n and ∑T≠0 for any non-empty subset T⊂D of cardinality |T|<n=12|D|. Also we prove that every finite decomposable subset D of an Abelian group contains two non-empty subsets A,B such that ∑A+∑B=0.

RELATIVE NON-COMMUTING GRAPH OF A FINITE GROUP

2012 ◽  
Vol 12 (02) ◽  
pp. 1250157 ◽  
Author(s):  
B. TOLUE ◽  
A. ERFANIAN
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
The Other ◽  
Commuting Graph ◽  
Other Hand ◽  
A Finite Group

The essence of the non-commuting graph remind us to find a connection between this graph and the commutativity degree as denoted by d(G). On the other hand, d(H, G) the relative commutativity degree, was the key to generalize the non-commuting graph ΓG to the relative non-commuting graph (denoted by ΓH, G) for a non-abelian group G and a subgroup H of G. In this paper, we give some results about ΓH, G which are mostly new. Furthermore, we prove that if (H1, G1) and (H2, G2) are relative isoclinic then ΓH1, G1 ≅ Γ H2, G2 under special conditions.

scholarly journals Maximal perpendicularity in certain Abelian groups

2017 ◽  
Vol 9 (1) ◽  
pp. 235-247
Author(s):  
Mika Mattila ◽  
Jorma K. Merikoski ◽  
Pentti Haukkanen ◽  
Timo Tossavainen
Keyword(s):  
Abelian Group ◽  
Vector Space ◽  
Binary Relation ◽  
Abelian Groups ◽  
The Other ◽  
Finitely Generated ◽  
Other Hand

AbstractWe define perpendicularity in an Abelian group G as a binary relation satisfying certain five axioms. Such a relation is maximal if it is not a subrelation of any other perpendicularity in G. A motivation for the study is that the poset (𝒫, ⊆) of all perpendicularities in G is a lattice if G has a unique maximal perpendicularity, and only a meet-semilattice if not. We study the cardinality of the set of maximal perpendicularities and, on the other hand, conditions on the existence of a unique maximal perpendicularity in the following cases: G ≅ ℤn, G is finite, G is finitely generated, and G = ℤ ⊕ ℤ ⊕ ⋯. A few such conditions are found and a few conjectured. In studying ℝn, we encounter perpendicularity in a vector space.

Random sets for the pointwise ergodic theorem

1992 ◽  
Vol 12 (1) ◽  
pp. 85-94 ◽  
Author(s):  
Yenkun Huang
Keyword(s):  
Ergodic Theorem ◽  
The Other ◽  
Random Sets ◽  
Simple Condition ◽  
Random Set ◽  
Large Classes ◽  
Pointwise Ergodic Theorem ◽  
Other Hand ◽  
Density Zero ◽  
Banach Density

AbstractWe generalize a result of Bourgain and devise more general criteria which guarantee that the corresponding random set in Z+ almost surely satisfies a pointwise ergodic theorem on Lp for p > 1. Several large classes of examples are constructed. We also show that under a simple condition the corresponding random set in Z+ almost surely satisfies a pointwise ergodic theorem not only on Lp for p > 1 but also on L1. On the other hand, we establish a criterion to conclude that a certain class of random sets have Banach density zero. In particular, all of the examples mentioned have Banach density zero.

scholarly journals On the Size of Dissociated Bases

10.37236/604 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Vsevolod F. Lev ◽  
Raphael Yuster
Keyword(s):  
Abelian Group ◽  
Finite Subset ◽  
Standard Basis ◽  
The Other ◽  
Logarithmic Factor ◽  
Other Hand

We prove that the sizes of the maximal dissociated subsets of a given finite subset of an abelian group differ by a logarithmic factor at most. On the other hand, we show that the set $\{0,1\}^n\subseteq\mathbb{Z}^n$ possesses a dissociated subset of size $\Omega(n\log n)$; since the standard basis of $\mathbb{Z}^n$ is a maximal dissociated subset of $\{0,1\}^n$ of size $n$, the result just mentioned is essentially sharp.

scholarly journals The free product of skew fields

1973 ◽  
Vol 16 (3) ◽  
pp. 300-308 ◽  
Author(s):  
P. M. Cohn
Keyword(s):  
Group Theory ◽  
Free Product ◽  
The Other ◽  
Group Case ◽  
Other Hand ◽  
Skew Fields

In a recent paper [3] it was shown that the free product K * L of two fields (possibly skew) can be embedded in a field, and moreover, this latter can be chosen to be the ‘universal field of fractions’ of K*L (cf. [4, 5]). This opens up the prospect of doing for skew fields what the Neumanns and others have done for groups; indeed some sample applications were given in [3]. We pursue this topic here a little further: our main results state (i) every ccuntably generated field can be embedded in a 2-generator field, (ii) in a free product of rings over a field k, any element algebraic over k is conjugate to an element in one of the factors, (iii) any field can be embedded in a field with nth roots for each n. These results are all analogous to well known results in group theory (cf. [8]), and although the proofs are not just a translation of the group case, the latter is of scme help. Thus (ii), (iii) follow fairly easily, but they lead to other problems, still open, by replacing ‘free product of rings’ in (ii) by ‘field product of fields, and in (iii) replacing ‘nth roots’ by ‘roots of any equation’. On the other hand, (i) is less immediate, since ‘field products’ need to be used in the proof, and their manipulation requires some more technical lemmas.

scholarly journals General Chemistry: Large Classes, Mixed Public, Three Languages (A Personal Experience)

2021 ◽  
Vol 75 (1) ◽  
pp. 39-44
Author(s):  
Katharina M. Fromm
Keyword(s):  
General Chemistry ◽  
Personal Experience ◽  
The Other ◽  
Large Classes ◽  
Future Studies ◽  
Other Hand ◽  
The University ◽  
Basic Level

At most universities, teaching general chemistry to fresh(wo)men is a challenge as the audience is usually composed of students of different backgrounds and interests. On one hand, the lecture is meant to bring all students to a basic level of chemistry required for future studies, on the other hand, certain concepts are discussed in much more depth than what students know from school. While it is already a balancing act to teach the content to students with little chemistry knowledge without boring those who took intensive classes at school, the University of Fribourg adds a challenge by teaching officially bilingual classes.

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