A Generalization of Groups

2020 ◽  
pp. 1-12
Author(s):  
Zekiye Çiloğlu Şahin ◽  
Yılmaz Çeven
Keyword(s):  
Group Theory ◽  
Binary Operation ◽  
Partial Group ◽  
Group A

For a nonempty set G, the authors define an operation * reckoned with closeness property (i.e.,* is an operation which is not a binary operation). Then they define the partial group as a generalisation of a group. A partial group G which is a non empty set satisfies following conditions hold for all a,b and c∈d:(PG1) If ab,(ab)c, bc and a (bc) is defined, then,(ab) c=a (bc)(PG2) . For every,a∈G there exists an e∈G such that ae and ea are defined and, ae=ea=a (PG3) . For every,a∈G there exists an a∈G such that aa and aa are defined and aa=aa=e. The * operation effects and changes the properties of group axioms. So that lots of group theoretic theorems and conclusions do not work in partial groups. Thus, this description gives us some fundemental and important properties and analogous to group theory. Also the authors have some differences from group theory.

‘Where chaos was, there shall be a matrix’ (de Maré). The large analytic group between chaos and creativity—reflections on the concept, its history and further evolution

2020 ◽  
pp. 053331642094267
Author(s):  
Peter Potthoff
Keyword(s):  
Group Theory ◽  
Small Group ◽  
Dyadic Relationship ◽  
Analytic Group ◽  
Large Group ◽  
Intervention Technique ◽  
Analytic Process ◽  
Group Experiences ◽  
Group A ◽  
Analytic Attitude

The author presents a survey of the development of large group theory. Older publications (e.g. Kreeger, Turqet, also drawing on Freud, M. Klein, Bion) describe the chaotic-aggressive, near-psychosis character of the large group, a certain contrast between the seemingly ‘benign’ small group and the ‘destructive’ large group and a predominantly dyadic relationship between the conductor and the large group. More recent publications (Wilke, Island, formerly also de Maré) underline the creative-constructive potential of the large group and the intersubjective interweaving of the conductor and the large group. Experiences with a conductor-pair in the daily large group of the Altaussee workshop instead of one conductor are discussed. The author pleads for the application of modern intersubjective theorizing to the large group. The analytic attitude and intervention technique would be changed: the traditional position of strict neutrality, anonymity and abstinence as well as the emphasis on whole-group interpretations would be substituted by a more open stance that does not hide the subjectivity of the conductor(s) but rather reveals and uses the conductor’s subjectivity to promote the analytic process.

PARTIAL ACTIONS OF GROUPS

2004 ◽  
Vol 14 (01) ◽  
pp. 87-114 ◽  
Author(s):  
J. KELLENDONK ◽  
MARK V. LAWSON
Keyword(s):  
Group Theory ◽  
Inverse Semigroup ◽  
Group Action ◽  
Partial Function ◽  
Inverse Semigroups ◽  
Explicit Description ◽  
Partial Action ◽  
Partial Group

A partial action of a group G on a set X is a weakening of the usual notion of a group action: the function G×X→X that defines a group action is replaced by a partial function; in addition, the existence of g·(h·x) implies the existence of (gh)·x, but not necessarily conversely. Such partial actions are extremely widespread in mathematics, and the main aim of this paper is to prove two basic results concerning them. First, we obtain an explicit description of Exel's universal inverse semigroup [Formula: see text], which has the property that partial actions of the group G give rise to actions of the inverse semigroup [Formula: see text]. We apply this result to the theory of graph immersions. Second, we prove that each partial group action is the restriction of a universal global group action. We describe some applications of this result to group theory and the theory of E-unitary inverse semigroups.

scholarly journals Partial associativity and rough approximate groups

2020 ◽  
Vol 30 (6) ◽  
pp. 1583-1647
Author(s):  
W. T. Gowers ◽  
J. Long
Keyword(s):  
Recent Result ◽  
Binary Operation ◽  
Multiplication Table ◽  
Additive Combinatorics ◽  
Positive Proportion ◽  
Finite Set ◽  
Group A ◽  
Metric Approximation ◽  
Metric Group

AbstractSuppose that a binary operation $$\circ $$ ∘ on a finite set X is injective in each variable separately and also associative. It is easy to prove that $$(X,\circ )$$ ( X , ∘ ) must be a group. In this paper we examine what happens if one knows only that a positive proportion of the triples $$(x,y,z)\in X^3$$ ( x , y , z ) ∈ X 3 satisfy the equation $$x\circ (y\circ z)=(x\circ y)\circ z$$ x ∘ ( y ∘ z ) = ( x ∘ y ) ∘ z . Other results in additive combinatorics would lead one to expect that there must be an underlying ‘group-like’ structure that is responsible for the large number of associative triples. We prove that this is indeed the case: there must be a proportional-sized subset of the multiplication table that approximately agrees with part of the multiplication table of a metric group. A recent result of Green shows that this metric approximation is necessary: it is not always possible to obtain a proportional-sized subset that agrees with part of the multiplication table of a group.

The need for closure

2019 ◽  
Vol 103 (557) ◽  
pp. 248-256
Author(s):  
Christopher D. Hollings
Keyword(s):  
Twentieth Century ◽  
Group Theory ◽  
Historical Perspective ◽  
Binary Operation ◽  
Need For Closure ◽  
Inherent Property

When defining a group, do we need to include closure? This is a detail that is often touched upon when the notion of a group is introduced to undergraduates. Should closure be listed as an axiom in its own right, or should it be regarded as an inherent property of the binary operation? There is no clear answer to this question, although there are firm opinions on both sides. Indeed, a very brief survey of group theory textbooks found in [1, pp. 458-459] suggests that there is a rough 50 : 50 split between authors who include closure explicitly and those who do not. In this Article, we go back to the beginning of the twentieth century to provide some historical perspective on this problem.

Group theory students’ perceptions of binary operation

2019 ◽  
Vol 103 (1) ◽  
pp. 63-81
Author(s):  
Kathleen Melhuish ◽  
Brittney Ellis ◽  
Michael D. Hicks
Keyword(s):  
Group Theory ◽  
Binary Operation

Centraliser rings of multiplication groups on quasigroups

1976 ◽  
Vol 79 (3) ◽  
pp. 427-431 ◽  
Author(s):  
J. D. H. Smith
Keyword(s):  
Group Theory ◽  
Vector Space ◽  
Symmetric Group ◽  
Binary Operation ◽  
Schur Ring ◽  
The Third ◽  
Permutation Matrices ◽  
Finite Set ◽  
Ring Method

Let (Q,.) be a finite quasigroup, i.e. a finite set Q with a binary operation. called multiplication such that in the equation x.y = z any two elements determine the third uniquely. Then the mappings R(x): Q → Q; q ↦ q.x and L(x): Q → Q; q ↦ x.q are permutations of Q. The multiplication group G of Q is the subgroup of the symmetric group on Q generated by {R(x), L(x) | x ∈ Q}. If S is a field, G has a faithful representation Ḡ by permutation matrices acting on the S-vector space with Q as basis. The set of matrices commuting with Ḡ forms an S-algebra (under the usual operations) called the centraliser ring V(G, Q) of G on Q. The purpose of this note is to show how the permutation-theoretic object ‘centraliser ring’ may be expressed in terms of the quasigroup structure of Q, both to prepare one tool for the long-term programme of classifying finite quasigroups by means of their multiplication groups, and for comparison with the Schur ring method of group theory.

scholarly journals Towards the Unification of Quantum Dynamics, Relativity and Living Organisms

2018 ◽  
Author(s):  
Arturo Tozzi ◽  
James F. Peters ◽  
John S. Torday
Keyword(s):  
General Relativity ◽  
Quantum Mechanics ◽  
Group Theory ◽  
Quantum Dynamics ◽  
Quantum Physics ◽  
Binary Operation ◽  
Relativity Theory ◽  
Mathematical Framework ◽  
Mathematical Approach ◽  
Living Organisms

The unexploited unification of quantum physics, general relativity and biology is a keystone that paves the way towards a better understanding of the whole of Nature.  Here we propose a mathematical approach that introduces the problem in terms of group theory.  We build a cyclic groupoid (a nonempty set with a binary operation defined on it) that encompasses the three frameworks as subsets, representing two of their most dissimilar experimental results, i.e., 1) the commutativity detectable both in our macroscopic relativistic world and in biology; 2) and the noncommutativity detectable both in the microscopic quantum world and in biology.  This approach leads to a mathematical framework useful in the investigation of the three apparently irreconcilable realms.  Also, we show how cyclic groupoids encompassing quantum mechanics, relativity theory and biology might be equipped with dynamics that can be described by paths on the twisted cylinder of a Möbius strip.   

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.

scholarly journals Splenic flexure mobilization is an essential role in laparoscopic low anterior resection!

2019 ◽  
Vol 6 (12) ◽  
pp. 4210
Author(s):  
Mohamed Hamed Elmeligi ◽  
Mohamed Sabry Amar ◽  
Mohammed Nazeeh Shaker Nassar
Keyword(s):  
Anterior Resection ◽  
Splenic Flexure ◽  
Conversion Rate ◽  
Low Anterior Resection ◽  
Operative Time ◽  
Splenic Flexure Mobilization ◽  
Partial Group ◽  
Laparoscopic Low Anterior Resection ◽  
Group A ◽  
Group B

Background: Routine mobilization of splenic flexure whether partial or complete became an essential step in laparoscopic low anterior resections in order to perform an oncologic re­section and to achieve a safe, tension-free anastomosis.Methods: 60 patients with rectal cancer were operated by laparoscopic low anterior resection with high ligation of inferior mesenteric artery in general surgery department, Menoufia university hospital between February 2016 and January 2019. All patients were divided randomly into 2 equal groups based on the techniques used in splenic flexure mobilization whether partial (group A) or complete (group B).Results: The majority of our patients were male 56.6% and 60% in both groups respectively with mean age (54.6±8.8) years in group A and mean age (58.5±9.2) years in group B. The operative time was highly significant lower in group A (269±17.6 minutes) than group B (304±22.4 minutes) while the conversion rate was significantly higher in group B (26.6%) than group A (6.6%). Regarding the postoperative data there was only significantly higher leak from the anastomosis in group A (20%) than group B (3.3%).Conclusions: Complete splenic flexure offer better oncological outcome and low incidence of anastomotic leak but with higher conversion rate, prolonged operative time, more blood loss and more 30 day mortality rate. So it needs more time to gain more experience to overcome these disadvantages.

scholarly journals On normally embedded subgroups of locally soluble FC-groups

1986 ◽  
Vol 28 (2) ◽  
pp. 153-159 ◽  
Author(s):  
J. C. Beidleman ◽  
M. J. Karbe
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Normal Subgroup ◽  
Sylow Subgroup ◽  
Infinite Group ◽  
Considerable Importance ◽  
Soluble Groups ◽  
Sylow Subgroups ◽  
Group A ◽  
A Finite Group

In his Habilitationsschrift [3] B. Fischer introduced the concept of a normally embedded subgroup of a finite group. A subgroup of a finite group G is said to be normally embedded in G if each of its Sylow subgroups is a Sylow subgroup of a normal subgroup of G. Meanwhile this concept has become of considerable importance in the theory of finite soluble groups and has been studied by various authors. However, in infinite group theory, normally embedded subgroups seem to have received little attention. The object of this note is to study normally embedded subgroups of locally soluble FC-groups.

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