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 Regularity and Green's Relations on a Semigroup of Transformations with Restricted Range

2008 ◽  
Vol 2008 ◽  
pp. 1-11 ◽  
Author(s):  
Jintana Sanwong ◽  
Worachead Sommanee
Keyword(s):  
Nonempty Subset ◽  
Transformation Semigroup ◽  
Green’S Relations ◽  
Sufficient Condition ◽  
Restricted Range ◽  
Necessary And Sufficient Condition ◽  
Full Transformation ◽  
Green's Relations ◽  
Necessary And Sufficient ◽  
Semigroup Of Transformations

LetT(X)be the full transformation semigroup on the setXand letT(X,Y)={α∈T(X):Xα⊆Y}. ThenT(X,Y)is a sub-semigroup ofT(X)determined by a nonempty subsetYofX. In this paper, we give a necessary and sufficient condition forT(X,Y)to be regular. In the case thatT(X,Y)is not regular, the largest regular sub-semigroup is obtained and this sub-semigroup is shown to determine the Green's relations onT(X,Y). Also, a class of maximal inverse sub-semigroups ofT(X,Y)is obtained.

REGULARITY AND STRONG REGULARITY IN THE CONTEXT OF CERTAIN CLASSES OF RINGS

2013 ◽  
Vol 12 (05) ◽  
pp. 1250205 ◽  
Author(s):  
MICHAŁ ZIEMBOWSKI
Keyword(s):  
Positive Integer ◽  
Sufficient Condition ◽  
Strong Regularity ◽  
Necessary And Sufficient Condition ◽  
Weak Dimension ◽  
Ring Of Polynomials ◽  
Necessary And Sufficient ◽  
Semihereditary Rings ◽  
The Ideal

We consider the ring R[x]/(xn+1), where R is a ring, R[x] is the ring of polynomials in an indeterminant x, (xn+1) is the ideal of R[x] generated by xn+1 and n is a positive integer. The aim of this paper is to show that regularity or strong regularity of a ring R is necessary and sufficient condition under which the ring R[x]/(xn+1) is an example of a ring which belongs to some important classes of rings. In this context, we discuss distributive rings, Bézout rings, Gaussian rings, quasi-morphic rings, semihereditary rings, and rings which have weak dimension less than or equal to one.

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.

scholarly journals A Construction in General Radical Theory

1970 ◽  
Vol 22 (6) ◽  
pp. 1097-1100 ◽  
Author(s):  
Augusto H. Ortiz
Keyword(s):  
Associative Ring ◽  
Semigroup Ring ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Radical Theory ◽  
Special Radical ◽  
Radical Property ◽  
Image Position ◽  
Necessary And Sufficient ◽  
The Ideal

Given an arbitrary associative ring R we consider the ring R[x] of polynomials over R in the commutative indeterminate x. For each radical property S we define the function S* which assigns to each ring R the idealof R. It is shown that the property SA (that a ring R be equal to S*(R)) is a radical property. If S is semiprime, then SA is semiprime also. If S is a special radical, then SA is a special radical. SA is always contained in S. A necessary and sufficient condition that S and SA coincide is given.The results are generalized in the last section to include extensions of R other than R[x], One such extension is the semigroup ring R[A], where A is a semigroup with an identity adjoined.

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 Cyclic partitions of complete nonuniform hypergraphs and complete multipartite hypergraphs

2013 ◽  
Vol Vol. 15 no. 2 (Combinatorics) ◽  
Author(s):  
Shonda Gosselin ◽  
Andrzej Szymański ◽  
Adam Pawel Wojda
Keyword(s):  
Positive Integer ◽  
Nonempty Subset ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Uniform Hypergraph ◽  
Necessary And Sufficient Condition ◽  
Vertex Set ◽  
International Audience ◽  
Complete Multipartite ◽  
Necessary And Sufficient

Combinatorics International audience A \em cyclic q-partition of a hypergraph (V,E) is a partition of the edge set E of the form \F,F^θ,F^θ², \ldots, F^θ^q-1\ for some permutation θ of the vertex set V. Let Vₙ = \ 1,2,\ldots,n\. For a positive integer k, Vₙ\choose k denotes the set of all k-subsets of Vₙ. For a nonempty subset K of V_n-1, we let \mathcalKₙ^(K) denote the hypergraph ≤ft(Vₙ, \bigcup_k∈ K Vₙ\choose k\right). In this paper, we find a necessary and sufficient condition on n, q and k for the existence of a cyclic q-partition of \mathcalKₙ^(V_k). In particular, we prove that if p is prime then there is a cyclic p^α-partition of \mathcalK^(Vₖ)ₙ if and only if p^α + β divides n, where β = \lfloor \logₚ k\rfloor. As an application of this result, we obtain two sufficient conditions on n₁,n₂,\ldots,n_t, k, α and a prime p for the existence of a cyclic p^α-partition of the complete t-partite k-uniform hypergraph \mathcal K^(k)_n₁,n₂,\ldots,n_t.

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 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 The Ideal Structures of Crossed Products of Cuntz Algebras by Quasi-Free Actions of Abelian Groups

2003 ◽  
Vol 55 (6) ◽  
pp. 1302-1338 ◽  
Author(s):  
Takeshi Katsura
Keyword(s):  
Abelian Groups ◽  
Sufficient Condition ◽  
Crossed Products ◽  
Necessary And Sufficient Condition ◽  
Cuntz Algebras ◽  
Necessary And Sufficient ◽  
Free Actions ◽  
The Ideal

AbstractWe completely determine the ideal structures of the crossed products of Cuntz algebras by quasi-free actions of abelian groups and give another proof of A. Kishimoto's result on the simplicity of such crossed products. We also give a necessary and sufficient condition that our algebras become primitive, and compute the Connes spectra andK-groups of our algebras.

TL-Strongly Implicative Ideals of BCI-Algebras

2013 ◽  
Vol 321-324 ◽  
pp. 2406-2409
Author(s):  
Xiao Zhang ◽  
Shao Quan Sun
Keyword(s):  
Algebra Homomorphism ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Brouwerian Lattice ◽  
Necessary And Sufficient

In this paper, the concepts of TL-strongly implicative ideal of BCI-algebra is introduced. A necessary and sufficient condition for a L- subset of BCI-algebra to be a L-strongly implicative ideal is stated, and inverse-images of TL-strongly implicative ideal under BCI-algebra homomorphism are studied. Also, several characterizations of TL-strongly implicative ideal are given. Where T is an arbitrary t-norm on a given complete Brouwerian lattice L.

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