scholarly journals Integral Cayley Multigraphs over Abelian and Hamiltonian Groups

10.37236/2742 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Matt DeVos ◽  
Roi Krakovski ◽  
Bojan Mohar ◽  
Azhvan Sheikh Ahmady
Keyword(s):  
Boolean Algebra ◽  
Character Sums ◽  
Sufficient Condition ◽  
Normal Subgroups ◽  
Necessary And Sufficient Condition ◽  
Generating Set ◽  
Theoretic Proof ◽  
Necessary And Sufficient ◽  
Hamiltonian Groups ◽  
Integral Cone

It is shown that a Cayley multigraph over a group $G$ with generating multiset $S$ is integral (i.e., all of its eigenvalues are integers) if $S$ lies in the integral cone over the boolean algebra generated by the normal subgroups of $G$. The converse holds in the case when $G$ is abelian. This in particular gives an alternative, character theoretic proof of a theorem of Bridges and Mena (1982). We extend this result to provide a necessary and sufficient condition for a Cayley multigraph over a Hamiltonian group to be integral, in terms of character sums and the structure of the generating set.

Skew quasi cyclic codes over 𝔽q + v𝔽q

2019 ◽  
Vol 18 (04) ◽  
pp. 1950077 ◽  
Author(s):  
Mehmet Özen ◽  
N. Tuğba Özzaim ◽  
Halit İnce
Keyword(s):  
Cyclic Codes ◽  
Quantum Codes ◽  
Computer Search ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generating Set ◽  
Skew Cyclic Codes ◽  
Necessary And Sufficient ◽  
Quantum Parameters ◽  
Quasi Cyclic Codes

In this work, skew quasi cyclic codes over [Formula: see text], where [Formula: see text] are considered. The generating set for one generator skew quasi cyclic codes over [Formula: see text] is also determined. We discuss a sufficient condition for one generator skew quasi cyclic codes to be free. Furthermore, a BCH type bound is given for free one generator skew quasi cyclic codes. We investigate the dual of skew quasi cyclic codes over [Formula: see text]. We give a necessary and sufficient condition for skew cyclic codes over [Formula: see text] to contain its dual. Moreover, we construct quantum codes from skew cyclic codes over [Formula: see text]. By using computer search we give some examples about skew quasi cyclic codes and list some quantum parameters in the table.

scholarly journals Applications of Fox's derivation in determining the generators of a group

2000 ◽  
Vol 61 (1) ◽  
pp. 27-32
Author(s):  
Wan Lin
Keyword(s):  
Normal Subgroup ◽  
Free Group ◽  
Quotient Group ◽  
Sufficient Conditions ◽  
Inverse Function ◽  
Sufficient Condition ◽  
Inverse Function Theorem ◽  
Necessary And Sufficient Condition ◽  
Generating Set ◽  
Necessary And Sufficient

We give a necessary and sufficient condition for a set of elements to be a generating set of a quotient group F/N, where F is the free group of rank n and N is a normal subgroup of F. Birman's Inverse Function Theorem is a corollary of our criterion. As an application of this criterion, we give necessary and sufficient conditions for a set of elements of the Burnside group B (n,p) of exponent p and rank n to be a generating set.

A class of sub-almost distributive lattices in an associate almost distributive lattice through a filter

2019 ◽  
Vol 13 (07) ◽  
pp. 2050136
Author(s):  
S. Ramesh ◽  
Jogarao Gunda
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Distributive Lattices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Algebraic Properties ◽  
Necessary And Sufficient

In this paper, we introduce a class of sub-almost distributive lattices in an associate almost distributive lattice through a filter. We obtain several algebraic properties on the class of sub-almost distributive lattices and prove that the above class forms a distributive lattice. We derive a necessary and sufficient condition that the class to become a Boolean algebra.

scholarly journals SOME REMARKS ON TOPOLOGICAL FULL GROUPS OF CANTOR MINIMAL SYSTEMS

2006 ◽  
Vol 17 (02) ◽  
pp. 231-251 ◽  
Author(s):  
HIROKI MATUI
Keyword(s):  
Sufficient Condition ◽  
Full Group ◽  
Normal Subgroups ◽  
Necessary And Sufficient Condition ◽  
Finitely Generated ◽  
Necessary And Sufficient ◽  
Cantor Minimal Systems

Giordano, Putnam and Skau showed that topological full groups of Cantor minimal systems are complete invariants for flip conjugacy. We will completely determine the structure of normal subgroups of the topological full group. Moreover, a necessary and sufficient condition for the topological full group to be finitely generated will be given.

Dense elements characterization of quasi-complemented Almost Distributive Lattices

2014 ◽  
Vol 07 (04) ◽  
pp. 1450062
Author(s):  
G. C. Rao ◽  
G. Nanaji Rao ◽  
A. Lakshmana
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

The properties of the set D of dense elements of an ADL are studied. The filter congruence θD generated by D in quasi-complemented ADLs is characterized. Quasi-complemented ADLs is characterized in terms of dense elements. A necessary and sufficient condition for a quasi-complemented ADLs to become a Boolean algebra is established.

scholarly journals On annihilators in BL-algebras

2016 ◽  
Vol 14 (1) ◽  
pp. 324-337 ◽  
Author(s):  
Yu Xi Zou ◽  
Xiao Long Xin ◽  
Peng Fei He
Keyword(s):  
Boolean Algebra ◽  
Nonempty Subset ◽  
Sufficient Condition ◽  
Ideal Lattice ◽  
Necessary And Sufficient Condition ◽  
Finite Product ◽  
Brouwerian Lattice ◽  
Necessary And Sufficient ◽  
Complemented Lattice ◽  
The Ideal

AbstractIn the paper, we introduce the notion of annihilators in BL-algebras and investigate some related properties of them. We get that the ideal lattice (I(L), ⊆) is pseudo-complemented, and for any ideal I, its pseudo-complement is the annihilator I⊥ of I. Also, we define the An (L) to be the set of all annihilators of L, then we have that (An(L); ⋂,∧An(L),⊥,{0}, L) is a Boolean algebra. In addition, we introduce the annihilators of a nonempty subset X of L with respect to an ideal I and study some properties of them. As an application, we show that if I and J are ideals in a BL-algebra L, then $J_I^ \bot $ is the relative pseudo-complement of J with respect to I in the ideal lattice (I(L), ⊆). Moreover, we get some properties of the homomorphism image of annihilators, and also give the necessary and sufficient condition of the homomorphism image and the homomorphism pre-image of an annihilator to be an annihilator. Finally, we introduce the notion of α-ideal and give a notation E(I ). We show that (E(I(L)), ∧E, ∨E, E(0), E(L) is a pseudo-complemented lattice, a complete Brouwerian lattice and an algebraic lattice, when L is a BL-chain or a finite product of BL-chains.

scholarly journals A Note on Boolean Like Algebras

2018 ◽  
Vol 7 (4.10) ◽  
pp. 1015
Author(s):  
K. Pushpalatha ◽  
V. M.L.Hima Bindu
Keyword(s):  
Boolean Algebra ◽  
Sufficient Condition ◽  
Boolean Ring ◽  
Necessary And Sufficient Condition ◽  
Abstract System ◽  
Binary Operations ◽  
Necessary And Sufficient

In this paper we develop on abstract system: viz Boolean-like algebra and prove that every Boolean  algebra is a Boolean-like algebra.  A necessary and sufficient condition for a Boolean-like algebra to be a Boolean algebra has been obtained.  As in the case of Boolean ring  and Boolean algebra, it is established that under suitable binary operations the Boolean-like ring and Boolean-like algebra are equivalent abstract structures. 

scholarly journals Relation Between Groups with Basis Property and Groups with Exchange Property

2016 ◽  
Vol 24 (2) ◽  
pp. 5-14
Author(s):  
Al Khalaf Khalaf ◽  
Mohammed Alkadhi
Keyword(s):  
Prime Order ◽  
Basis Property ◽  
Polycyclic Group ◽  
Exchange Property ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generating Set ◽  
Necessary And Sufficient ◽  
Minimal Generating Set

AbstractA group G is called a group with basis property if there exists a basis (minimal generating set) for every subgroup H of G and every two bases are equivalent. A group G is called a group with exchange property, if x∉〈X〉 ⋀ x∈〈X∪{y}〉, then y∈〈X∪{x}〉, for all x, y ∈ G and for every subset X⊆G. In this research, we proved the following: Every polycyclic group satisfies the basis property. Every element in a group with the exchange property has a prime order. Every p-group satisfies the exchange property if and only if it is an elementary abelian p-group. Finally, we found necessary and sufficient condition for every group to satisfy the exchange property, based on a group with the basis property.

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