Semigroup Structures and Commutative Ideals of BCK-Algebras Based on Crossing Cubic Set Structures

First, semigroup structure is constructed by providing binary operations for the crossing cubic set structure. The concept of commutative crossing cubic ideal is introduced by applying crossing cubic set structure to commutative ideal in BCK-algebra, and several properties are investigated. The relationship between crossing cubic ideal and commutative crossing cubic ideal is discussed. An example to show that crossing cubic ideal is not commutative crossing cubic ideal is given, and then the conditions in which crossing cubic ideal can be commutative crossing cubic ideal are explored. Characterizations of commutative crossing cubic ideal are discussed, and the relationship between commutative crossing cubic ideal and crossing cubic level set is considered. An extension property of commutative crossing cubic ideal is established, and the translation of commutative crossing cubic ideal is studied. Conditions for the translation of crossing cubic set structure to be commutative crossing cubic ideal are provided, and its characterization is processed.

Extension Property of Automorphism of Modules

In this paper we generalize Schupp’s result for groups to modules. For an injective module, every automorphism satisfies the extension property. We characterize the automorphisms of a module M satisfies the extension property.

On local spectral properties of operator matrices

In this paper, we focus on a $2 \times 2$ 2 × 2 operator matrix $T_{\epsilon _{k}}$ T ϵ k as follows: $$\begin{aligned} T_{\epsilon _{k}}= \begin{pmatrix} A & C \\ \epsilon _{k} D & B\end{pmatrix}, \end{aligned}$$ T ϵ k = ( A C ϵ k D B ) , where $\epsilon _{k}$ ϵ k is a positive sequence such that $\lim_{k\rightarrow \infty }\epsilon _{k}=0$ lim k → ∞ ϵ k = 0 . We first explore how $T_{\epsilon _{k}}$ T ϵ k has several local spectral properties such as the single-valued extension property, the property $(\beta )$ ( β ) , and decomposable. We next study the relationship between some spectra of $T_{\epsilon _{k}}$ T ϵ k and spectra of its diagonal entries, and find some hypotheses by which $T_{\epsilon _{k}}$ T ϵ k satisfies Weyl’s theorem and a-Weyl’s theorem. Finally, we give some conditions that such an operator matrix $T_{\epsilon _{k}}$ T ϵ k has a nontrivial hyperinvariant subspace.

Sobolev Extension on Lp-quasidisks

In this paper, we study the Sobolev extension property of Lp-quasidisks which are the generalizations of classical quasidisks. After that, we also find some applications of this property.

Quasi-N4-Lattices

Within the Nelson family, two mutually incomparable generalizations of Nelson constructive logic with strong negation have been proposed so far. The first and more well-known, Nelson paraconsistent logic , results from dropping the explosion axiom of Nelson logic; a more recent series of papers considers the logic (dubbed quasi-Nelson logic ) obtained by rejecting the double negation law, which is thus also weaker than intuitionistic logic. The algebraic counterparts of these logical calculi are the varieties of N4-lattices and quasi-Nelson algebras . In the present paper we propose the class of quasi- N4-lattices as a common generalization of both. We show that a number of key results, including the twist-structure representation of N4-lattices and quasi-Nelson algebras, can be uniformly established in this more general setting; our new representation employs twist-structures defined over Brouwerian algebras enriched with a nucleus operator. We further show that quasi-N4-lattices form a variety that is arithmetical, possesses a ternary as well as a quaternary deductive term, and enjoys EDPC and the strong congruence extension property.

Ideals of Residuated Lattices

In this article, we study ideals in residuated lattice and present a characterization theorem for them. We investigate some related results between the obstinate ideals and other types of ideals of a residuated lattice, likeness Boolean, primary, prime, implicative, maximal and ʘ-prime ideals. Characterization theorems and extension property for obstinate ideal are stated and proved. For the class of ʘ-residuated lattices, by using the ʘ-prime ideals we propose a characterization, and prove that an ideal is an ʘ-prime ideal iff its quotient algebra is an ʘ-residuated lattice. Finally, by using ideals, the class of Noetherian (Artinian) residuated lattices is introduced and Cohen’s theorem is proved.

Imploring GE-Filters of GE-Algebras

Relations between a transitive GE-algebra, a belligerent GE-algebra, an antisymmetric GE-algebra, and a left exchangeable GE-algebra are displayed. A new substructure, so called imploring GE-filter, is introduced, and its properties are investigated. The relationship between a GE-filter, an imploring GE-filter, a belligerent GE-filter, and a prominent GE-filter are considered. Conditions for an imploring GE-filter to be a belligerent GE-filter are given, and the conditions necessary for a (belligerent) GE-filter to be an imploring GE-filter are found. Relations between a prominent GE-filter and an imploring GE-filter are discussed, and a condition for an imploring GE-filter to be a prominent GE-filter is provided. Examples to show that a belligerent GE-filter and a prominent GE-filer are independent concepts are given. The extension property of the imploring GE-filter is established.

Bipolar Fuzzy Implicative Ideals of BCK-Algebras

The notion of bipolar fuzzy implicative ideals of a BCK-algebra is introduced, and several properties are investigated. The relation between a bipolar fuzzy ideal and a bipolar fuzzy implicative ideal is studied. Characterizations of a bipolar fuzzy implicative ideal are given. Conditions for a bipolar fuzzy set to be a bipolar fuzzy implicative ideal are provided. Extension property for a bipolar fuzzy implicative ideal is stated.