scholarly journals On the classification of Schreier extensions of monoids with non-abelian kernel

2020 ◽  
Vol 32 (3) ◽  
pp. 607-623
Author(s):  
Nelson Martins-Ferreira ◽  
Andrea Montoli ◽  
Alex Patchkoria ◽  
Manuela Sobral
Keyword(s):  
Abelian Group ◽  
Cohomology Group ◽  
Isomorphism Classes ◽  
The Third ◽  
Abelian Kernel ◽  
Invertible Elements ◽  
Abstract Kernel

AbstractWe show that any regular (right) Schreier extension of a monoid M by a monoid A induces an abstract kernel {\Phi\colon M\to\frac{\operatorname{End}(A)}{\operatorname{Inn}(A)}}. If an abstract kernel factors through {\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}}, where {\operatorname{SEnd}(A)} is the monoid of surjective endomorphisms of A, then we associate to it an obstruction, which is an element of the third cohomology group of M with coefficients in the abelian group {U(Z(A))} of invertible elements of the center {Z(A)} of A, on which M acts via Φ. An abstract kernel {\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} (resp. {\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{Inn}(A)}}) is induced by a regular weakly homogeneous (resp. homogeneous) Schreier extension of M by A if and only if its obstruction is zero. We also show that the set of isomorphism classes of regular weakly homogeneous (resp. homogeneous) Schreier extensions inducing a given abstract kernel {\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} (resp. {\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{Inn}(A)}}), when it is not empty, is in bijection with the second cohomology group of M with coefficients in {U(Z(A))}.

scholarly journals The third partial cohomology group and existence of extensions of semilattices of groups by groups

2020 ◽  
Vol 32 (5) ◽  
pp. 1297-1313
Author(s):  
Mikhailo Dokuchaev ◽  
Mykola Khrypchenko ◽  
Mayumi Makuta
Keyword(s):  
Cohomology Group ◽  
The Third ◽  
The Given ◽  
Abstract Kernel

AbstractWe introduce the concept of a partial abstract kernel associated to a group G and a semilattice of groups A and relate the partial cohomology group {H^{3}(G,C(A))} with the obstructions to the existence of admissible extensions of A by G which realize the given abstract kernel. We also show that if such extensions exist, then they are classified by {H^{2}(G,C(A))}.

CLASSIFICATION OF MINIMAL NONASSOCIATIVE MOUFANG LOOPS OF ORDER pq3

2013 ◽  
Vol 23 (08) ◽  
pp. 1895-1908 ◽  
Author(s):  
WING LOON CHEE ◽  
STEPHEN M. GAGOLA ◽  
ANDREW RAJAH
Keyword(s):  
Abelian Group ◽  
Open Problem ◽  
Related Question ◽  
Moufang Loop ◽  
The Third ◽  
Odd Order ◽  
Moufang Loops ◽  
Mod P

An open problem in the theory of Moufang loops is to classify those loops which are minimally nonassociative, that is, loops which are nonassociative but where all proper subloops are associative. A related question is to classify all integers n for which a minimally nonassociative loop exists. In [Possible orders of nonassociative Moufang loops, Comment. Math. Univ. Carolin.41(2) (2000) 237–244], O. Chein and the third author showed that a minimal nonassociative Moufang loop of order 2q3can be constructed by using a non-abelian group of order q3. In [Moufang loops of odd order pq3, J. Algebra235 (2001) 66–93], the third author also proved that for odd primes p < q, a nonassociative Moufang loop of order pq3exists if and only if q ≡ 1 ( mod p). Here we complete the classification of minimally nonassociative Moufang loops of order pq3for primes p < q.

scholarly journals Baer sums for a natural class of monoid extensions

2021 ◽  
Author(s):  
Peter F. Faul
Keyword(s):  
Cohomology Group ◽  
Natural Class ◽  
Isomorphism Classes ◽  
Full Characterization ◽  
Abelian Kernel ◽  
Split Extensions ◽  
Baer Sum ◽  
Suitable Notion ◽  
Factor Set

AbstractIt is well known that the set of isomorphism classes of extensions of groups with abelian kernel is characterized by the second cohomology group. In this paper we generalise this characterization of extensions to a natural class of extensions of monoids, the cosetal extensions. An extension "Equation missing" is cosetal if for all $$g,g' \in G$$ g , g ′ ∈ G in which $$e(g) = e(g')$$ e ( g ) = e ( g ′ ) , there exists a (not necessarily unique) $$n \in N$$ n ∈ N such that $$g = k(n)g'$$ g = k ( n ) g ′ . These extensions generalise the notion of special Schreier extensions, which are themselves examples of Schreier extensions. Just as in the group case where a semidirect product could be associated to each extension with abelian kernel, we show that to each cosetal extension (with abelian group kernel), we can uniquely associate a weakly Schreier split extension. The characterization of weakly Schreier split extensions is combined with a suitable notion of a factor set to provide a cohomology group granting a full characterization of cosetal extensions, as well as supplying a Baer sum.

ABOUT THE PROBLEM OF CLASSIFICATION OF THE CONCEPTS OF INFORMATICS STUDIED AT SECONDARY SCHOOL

2018 ◽  
pp. 4-7
Author(s):  
S. I. Zenko
Keyword(s):  
Secondary School ◽  
Foreign Language ◽  
Computer Science ◽  
Parts Of Speech ◽  
The Third ◽  
The Cross ◽  
Definition Of

The article raises the problem of classification of the concepts of computer science and informatics studied at secondary school. The efficiency of creation of techniques of training of pupils in these concepts depends on its solution. The author proposes to consider classifications of the concepts of school informatics from four positions: on the cross-subject basis, the content lines of the educational subject "Informatics", the logical and structural interrelations and interactions of the studied concepts, the etymology of foreign-language and translated words in the definition of the concepts of informatics. As a result of the first classification general and special concepts are allocated; the second classification — inter-content and intra-content concepts; the third classification — stable (steady), expanding, key and auxiliary concepts; the fourth classification — concepts-nouns, conceptsverbs, concepts-adjectives and concepts — combinations of parts of speech.

(A review). BRAUN-BLANQUETIA. Recueil de travaux de geobotanique / review of geobotanical monographs. Vol. 46. Centenaire de la phytosociologie. CAMERINO, 2010. 423 p.

2011 ◽  
pp. 143-147
Author(s):  
L. G. Naumova ◽  
V. B. Martynenko ◽  
S. M. Yamalov
Keyword(s):  
Plant Association ◽  
Northern Asia ◽  
The Third ◽  
International Botanical ◽  
International Botanical Congress ◽  
History Of ◽  
Definition Of

Date of «birth» of phytosociology (phytocenology) is considered to be 1910, when at the third International Botanical Congress in Brussels adopted the definition of plant association in the wording Including Flaó and K. Schröter (Flahault, Schröter, 1910; Alexandrov, 1969). The centenary of this momentous event in the history of phytocenology devoted to the 46th edition of the Yearbook «Braun-Blanquetia», which began to emerge in 1984 in Camerino (Italy) and it has a task to publish large geobotanical works. During the years of the publication of the Yearbook on its pages were published twice work of the Russian scientists — «The steppes of Mongolia» (Z. V. Karamysheva, V. N. Khramtsov. Vol. 17. 1995), and «Classification of continental hemiboreal forests of Northern Asia» (N. B. Ermakov in collaboration with English colleagues and J. Dring, J. Rodwell. Vol. 28. 2000).

scholarly journals ABELIAN GROUPS WITH A p2-BOUNDED SUBGROUP, REVISITED

2011 ◽  
Vol 10 (03) ◽  
pp. 377-389
Author(s):  
CARLA PETRORO ◽  
MARKUS SCHMIDMEIER
Keyword(s):  
Abelian Group ◽  
Prime Number ◽  
Maximal Ideal ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Uniserial Ring

Let Λ be a commutative local uniserial ring of length n, p be a generator of the maximal ideal, and k be the radical factor field. The pairs (B, A) where B is a finitely generated Λ-module and A ⊆B a submodule of B such that pmA = 0 form the objects in the category [Formula: see text]. We show that in case m = 2 the categories [Formula: see text] are in fact quite similar to each other: If also Δ is a commutative local uniserial ring of length n and with radical factor field k, then the categories [Formula: see text] and [Formula: see text] are equivalent for certain nilpotent categorical ideals [Formula: see text] and [Formula: see text]. As an application, we recover the known classification of all pairs (B, A) where B is a finitely generated abelian group and A ⊆ B a subgroup of B which is p2-bounded for a given prime number p.

To the study of early dementia

1902 ◽  
Vol 2 (9) ◽  
pp. 480-481
Author(s):  
V. Serbskiy
Keyword(s):  
Clinical Course ◽  
Mental Illnesses ◽  
Dementia Praecox ◽  
The Third ◽  
Current State ◽  
Early Dementia

In the first part of his article, the author examines the current state of the issue of secondary dementia and proves that a group of psychoses, known under the name secondary dementia, should be left in the classification of mental illnesses. The second part is devoted to the analysis of Krpelin's scholarship on dementia praecox, and the author fundamentally disagrees with many of the provisions of the latter. In the third part, the ethiology, clinical course and recognition of premature dementia are analyzed.

Latviešu metālmūzikas albumu nosaukumi

2021 ◽  
pp. 361-369
Author(s):  
Ingars Gusāns
Keyword(s):  
Current Situation ◽  
Creative Work ◽  
Distinctive Character ◽  
Literary Works ◽  
The Third ◽  
Metal Music ◽  
Local Character ◽  
The Subject ◽  
The Common

The aim of the study is to describe the titles of Latvian metal music albums, from the perspective of content, by identifying the common and distinctive character of the metallic music tradition, and perhaps even the local one. Of 241 album titles (data on Dec. 31, 2019), most are in English, some in French, Latin, Russian, some consisting of digits, and 69 titles in Latvian. These titles are the subject of the research. The main source is Encyclopaedia Metallum (www.metal-archives.com), which still does not reflect the current situation concerning Latvian metal music. Album titles in this study are viewed separately from album designs and song titles and are analysed from the perspective of content. The album title is an important part of the work that has been issued because it is an element that makes the audience/buyer pay attention to the album because it must not be forgotten that today the album is also an item that you want to sell. In general, it can be concluded that Latvian metal musicians, with their album titles in Latvian, are mostly following world trends, as evidenced by the integration in the researcher Deena Weinstein’s classification of Dionysian discourse and discourse on chaos. Most titles are more relevant to the discourse on chaos because the thematic circle of chaos is wider. Latvian mythology, along with history, is an up-to-date source for the creative work of bands that is responsible for the local feeling of the titles. A large enough number are titles that are difficult to fit in the Weinstein’s division and form the third group with philosophical titles and simply all sorts of titles. If the philosophical titles follow the world’s trends, the simple titles include the names of the events, tributes, and the titles of literary works, which give them a local character.

Classification of Au Myths

1972 ◽  
Vol os-19 (5) ◽  
pp. 214-218
Author(s):  
David Scorza
Keyword(s):  
Papua New Guinea ◽  
New Guinea ◽  
Older Men ◽  
The Third ◽  
Crosscultural Communication

Folklore among the Au of Papua New Guinea takes three forms: Tipiitim Tipiir, tales told the ancestors by the spirits; Tipiitim Herwe, tales invented by the ancestors; and Him, tales invented by the older men. The first are basically etiological, the second provide emotional release and entertainment, and the third show attempts to cope with the cultural flux which followed the arrival of Europeans. Certain Scriptural parallels are explored, with their implications for crosscultural communication of the gospel.

COMPOSITION SERIES OF THE SOLVABLE ABELIAN FACTOR GROUP SOURCE OF EQUATION OF ALL POLYNOMIAL EQUATIONS

2021 ◽  
Vol 5 (2) ◽  
pp. 462-469
Author(s):  
Bernard Alechenu ◽  
Babayo Muhammed Abdullahi ◽  
Daniel Eneojo Emmanuel
Keyword(s):  
Abelian Group ◽  
Complex Analysis ◽  
Factor Group ◽  
Roots Of Unity ◽  
Composition Series ◽  
Isomorphism Theorem ◽  
Polynomial Equations ◽  
Canonical Map ◽  
The Third ◽  
Recursive Set

This work penciled down the Composition Series of Factor Abelian Group over the source of all polynomial equations gleaned through  the nth roots of unity regular gons on a unit circle, a circle of radius one and centered at zero. To get the composition series, the third isomorphism theorem has to be passed through. But, the third isomorphism theorem itself gleaned via the first which is a deduction of the naturally existing canonical map. The solution of the source atom of the equation of all equation of polynomials are solvable by the intertwine of the Euler’s Formula and the De Moivre’s Theorem which after the inter-math, they become within the domain of complex analysis. For the source root of the equations, there is a recursive set of homomorphisms and ontoness of the mappings geneting the sequential terms in the composition series.    

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