CHARACTERIZING n-ISOCLINIC CLASSES OF CROSSED MODULES

2018 ◽  
Vol 61 (03) ◽  
pp. 637-656
Author(s):  
HAJAR RAVANBOD ◽  
ALI REZA SALEMKAR ◽  
SAJEDEH TALEBTASH
Keyword(s):  
Equivalence Relation ◽  
Crossed Module ◽  
Basic Properties ◽  
Crossed Modules

AbstractIn this paper, we introduce the notion of the equivalence relation, called n-isoclinism, between crossed modules of groups, and give some basic properties of this notion. In particular, we obtain some criteria under which crossed modules are n-isoclinic. Also, we present the notion of n-stem crossed module and, under some conditions, determine them within an n-isoclinism class.

scholarly journals From simplicial homotopy to crossed module homotopy in modified categories of interest

2020 ◽  
Vol 27 (4) ◽  
pp. 541-556
Author(s):  
Kadir Emir ◽  
Selim Çetin
Keyword(s):  
Lie Algebras ◽  
Equivalence Relation ◽  
Crossed Module ◽  
Algebraic Structures ◽  
Homotopy Equivalence ◽  
Associative Algebras ◽  
Leibniz Algebras ◽  
Crossed Modules ◽  
Simplicial Objects ◽  
Simplicial Homotopy

AbstractWe address the (pointed) homotopy of crossed module morphisms in modified categories of interest that unify the notions of groups and various algebraic structures. We prove that the homotopy relation gives rise to an equivalence relation as well as to a groupoid structure with no restriction on either domain or co-domain of the corresponding crossed module morphisms. Furthermore, we also consider particular cases such as crossed modules in the categories of associative algebras, Leibniz algebras, Lie algebras and dialgebras of the unified homotopy definition. Finally, as one of the major objectives of this paper, we prove that the functor from simplicial objects to crossed modules in modified categories of interest preserves the homotopy as well as the homotopy equivalence.

Isoclinism of crossed modules

2016 ◽  
Vol 09 (03) ◽  
pp. 1650091 ◽  
Author(s):  
Ali Reza Salemkar ◽  
Hamid Mohammadzadeh ◽  
Saeed Shahrokhi
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Crossed Module ◽  
Direct Products ◽  
Group Case ◽  
Crossed Modules ◽  
Equivalent Conditions

In this paper, we introduce the notion of the equivalence relation, isoclinism, on crossed modules of groups, and give some equivalent conditions for crossed modules to be isoclinic. In particular, it is shown that if two crossed modules [Formula: see text] and [Formula: see text] are isoclinic then [Formula: see text] can be constructed from [Formula: see text] and vice versa using the operations of forming direct products, taking crossed submodules, and factoring crossed modules, which generalizes the work of Weichsel. Also, similar to a result of Hall in the group case, we show that any equivalence class of crossed modules contains at least one stem crossed module.

Internal Crossed Modules

2003 ◽  
Vol 10 (1) ◽  
pp. 99-114 ◽  
Author(s):  
G. Janelidze
Keyword(s):  
Crossed Module ◽  
Crossed Modules

Abstract We introduce the notion of (pre)crossed module in a semiabelian category, and establish equivalences between internal reflexive graphs and precrossed modules, and between internal categories and crossed modules.

An axiomatization of the logic with the rough quantifier

1991 ◽  
Vol 56 (2) ◽  
pp. 608-617 ◽  
Author(s):  
Michał Krynicki ◽  
Hans-Peter Tuschik
Keyword(s):  
Equivalence Relation ◽  
Equivalence Relations ◽  
Order Structure ◽  
Partition Model ◽  
Weak Model ◽  
First Order ◽  
Basic Properties ◽  
Generalized Quantifier ◽  
Order Language ◽  
The Right

We consider the language L(Q), where L is a countable first-order language and Q is an additional generalized quantifier. A weak model for L(Q) is a pair 〈, q〉 where is a first-order structure for L and q is a family of subsets of its universe. In case that q is the set of classes of some equivalence relation the weak model 〈, q〉 is called a partition model. The interpretation of Q in partition models was studied by Szczerba [3], who was inspired by Pawlak's paper [2]. The corresponding set of tautologies in L(Q) is called rough logic. In the following we will give a set of axioms of rough logic and prove its completeness. Rough logic is designed for creating partition models.The partition models are the weak models arising from equivalence relations. For the basic properties of the logic of weak models the reader is referred to Keisler's paper [1]. In a weak model 〈, q〉 the formulas of L(Q) are interpreted as usual with the additional clause for the quantifier Q: 〈, q〉 ⊨ Qx φ(x) iff there is some X ∊ q such that 〈, q〉 ⊨ φ(a) for all a ∊ X.In case X satisfies the right side of the above equivalence we say that X is contained in φ(x) or, equivalently, φ(x) contains X.

scholarly journals Group cohomology with coefficients in a crossed module

2010 ◽  
Vol 10 (2) ◽  
pp. 359-404 ◽  
Author(s):  
Behrang Noohi
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Geometric Approach ◽  
Group Cohomology ◽  
Theoretic Approach ◽  
Crossed Module ◽  
Explicit Approach ◽  
Crossed Modules ◽  
Cohomology Sequence

AbstractWe compare three different ways of defining group cohomology with coefficients in a crossed module: (1) explicit approach via cocycles; (2) geometric approach via gerbes; (3) group theoretic approach via butterflies. We discuss the case where the crossed module is braided and the case where the braiding is symmetric. We prove the functoriality of the cohomologies with respect to weak morphisms of crossed modules and also prove the ‘long’ exact cohomology sequence associated to a short exact sequence of crossed modules and weak morphisms.

SEMI-COMPLETE CROSSED MODULES OF LIE ALGEBRAS

2012 ◽  
Vol 11 (05) ◽  
pp. 1250096
Author(s):  
A. AYTEKIN ◽  
J. M. CASAS ◽  
E. Ö. USLU
Keyword(s):  
Lie Algebras ◽  
Necessary Conditions ◽  
Crossed Module ◽  
Crossed Modules ◽  
Sufficient And Necessary Conditions

We investigate some sufficient and necessary conditions for (semi)-completeness of crossed modules in Lie algebras and we establish its relationships with the holomorphy of a crossed module. When we consider Lie algebras as crossed modules, then we recover the corresponding classical results for complete Lie algebras.

scholarly journals Quasirationality and prounipotent crossed modules

2019 ◽  
Vol 28 (13) ◽  
pp. 1940012
Author(s):  
A. M. Mikhovich
Keyword(s):  
Crossed Module ◽  
Defining Relation ◽  
Crossed Modules

We study quasirational (QR) presentations of (pro-[Formula: see text])groups, which contain aspherical presentations and their subpresentations, and also still mysterious pro-[Formula: see text]-groups with a single defining relation. Using schematization of QR-presentations and embedding of the rationalized module of relations into a diagram related to a certain prounipotent crossed module, we derive cohomological properties of pro-[Formula: see text]-groups with a single defining relation.

scholarly journals Anti Fuzzy Equivalence Relation on Rings with Respect to t-conorm C

2019 ◽  
pp. 1-19 ◽  
Author(s):  
Rasul Rasuli
Keyword(s):  
Equivalence Relation ◽  
Congruence Relation ◽  
Basic Properties ◽  
Fuzzy Congruence

In this paper, by using t-conorms, we define the concept of anti fuzzy equivalence relation and anti fuzzy congruence relation on ring R and we investigate some of their basic properties. Also we define fuzzy ideals of ring R under t-conorms and compare this with fuzzy equivalence relation and fuzzy congruence relation on ring R such that we define new introduced ring. Next we investigate this concept under homomorphism of new introduced ring.

scholarly journals Normality and quotient in the category of crossed modules within the category of groups with operations

2019 ◽  
Vol 38 (7) ◽  
pp. 169-179
Author(s):  
Tunçar Şahan ◽  
Osman Mucuk
Keyword(s):  
Crossed Module ◽  
Crossed Modules

In this paper we define the notions of normal subcrossed module and quotient crossed module within groups with operations; and then give same properties of such crossed modules in groups with operations.

scholarly journals Liftings of crossed modules in the category of groups with operations

2019 ◽  
Vol 38 (7) ◽  
pp. 181-193
Author(s):  
H. Fulya Akız ◽  
Osman Mucuk ◽  
Tunçar Şahan
Keyword(s):  
Crossed Module ◽  
Crossed Modules ◽  
Internal Groupoid

In this paper we define the notion of lifting of a crossed module via the morphism in groups with operations and give some properties of this type of liftings. Further we prove that the lifting crossed modules of a certain crossed module are categorically equivalent to the internal groupoid actions on groups with operations, where the internal groupoid corresponds to the crossed module.

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