scholarly journals Cancellation problem for AS-Regular algebras of dimension three

2021 ◽  
Vol 312 (1) ◽  
pp. 233-256
Author(s):  
Xin Tang ◽  
Helbert J. Venegas Ramírez ◽  
James J. Zhang
Keyword(s):  
Cancellation Problem ◽  
Regular Algebras

scholarly journals Dynamic Łukasiewicz logic and its application to immune system

2021 ◽  
Author(s):  
Antonio Di Nola ◽  
Revaz Grigolia ◽  
Nunu Mitskevich ◽  
Gaetano Vitale
Keyword(s):  
Immune System ◽  
Scalar Multiplication ◽  
Kripke Semantics ◽  
Łukasiewicz Logic ◽  
Lukasiewicz Logic ◽  
Regular Algebras

AbstractIt is introduced an immune dynamic n-valued Łukasiewicz logic $$ID{\L }_n$$ I D Ł n on the base of n-valued Łukasiewicz logic $${\L }_n$$ Ł n and corresponding to it immune dynamic $$MV_n$$ M V n -algebra ($$IDL_n$$ I D L n -algebra), $$1< n < \omega $$ 1 < n < ω , which are algebraic counterparts of the logic, that in turn represent two-sorted algebras $$(\mathcal {M}, \mathcal {R}, \Diamond )$$ ( M , R , ◊ ) that combine the varieties of $$MV_n$$ M V n -algebras $$\mathcal {M} = (M, \oplus , \odot , \sim , 0,1)$$ M = ( M , ⊕ , ⊙ , ∼ , 0 , 1 ) and regular algebras $$\mathcal {R} = (R,\cup , ;, ^*)$$ R = ( R , ∪ , ; , ∗ ) into a single finitely axiomatized variety resembling R-module with “scalar” multiplication $$\Diamond $$ ◊ . Kripke semantics is developed for immune dynamic Łukasiewicz logic $$ID{\L }_n$$ I D Ł n with application in immune system.

scholarly journals Neighborhood Reconstruction and Cancellation of Graphs

10.37236/6676 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Richard H. Hammack ◽  
Cristina Mullican
Keyword(s):  
Direct Product ◽  
Structural Characterization ◽  
Unsolved Problem ◽  
Reconstruction Problem ◽  
Cancellation Problem ◽  
Open Vertex ◽  
Isomorphic Graphs ◽  
Product Of Graphs

We connect two seemingly unrelated problems in graph theory.Any graph $G$ has a neighborhood multiset $\mathscr{N}(G)= \{N(x) \mid x\in V(G)\}$ whose elements are precisely the open vertex-neighborhoods of $G$. In general there exist non-isomorphic graphs $G$ and $H$ for which $\mathscr{N}(G)=\mathscr{N}(H)$. The neighborhood reconstruction problem asks the conditions under which $G$ is uniquely reconstructible from its neighborhood multiset, that is, the conditions under which $\mathscr{N}(G)=\mathscr{N}(H)$ implies $G\cong H$. Such a graph is said to be neighborhood-reconstructible.The cancellation problem for the direct product of graphs seeks the conditions under which $G\times K\cong H\times K$ implies $G\cong H$. Lovász proved that this is indeed the case if $K$ is not bipartite. A second instance of the cancellation problem asks for conditions on $G$ that assure $G\times K\cong H\times K$ implies $G\cong H$ for any bipartite~$K$ with $E(K)\neq \emptyset$. A graph $G$ for which this is true is called a cancellation graph.We prove that the neighborhood-reconstructible graphs are precisely the cancellation graphs. We also present some new results on cancellation graphs, which have corresponding implications for neighborhood reconstruction. We are particularly interested in the (yet-unsolved) problem of finding a simple structural characterization of cancellation graphs (equivalently, neighborhood-reconstructible graphs).

Artin–Schelter regular algebras of dimension five with three generators

2021 ◽  
pp. 2250235
Author(s):  
Jun Li
Keyword(s):  
Dimension Formula ◽  
Regular Algebras ◽  
Generating Relations ◽  
Degree Type

In this paper, we investigate Artin–Schelter regular algebras of dimension [Formula: see text] with three generators in degree [Formula: see text] under the hypothesis that [Formula: see text], in which the degree types of the relations for the number of the generating relations less than five can be determined. We prove that the only possible degree type of three generating relations is [Formula: see text] and the only possible degree type of four generating relations is [Formula: see text].

scholarly journals Regular algebras of dimension 4 and their $A_\infty$ -Ext-algebras

2007 ◽  
Vol 137 (3) ◽  
pp. 537-584 ◽  
Author(s):  
D.-M. Lu ◽  
J. H. Palmieri ◽  
Q.-S. Wu ◽  
J. J. Zhang
Keyword(s):  
Regular Algebras

HOMOGENEOUS PBW DEFORMATION FOR ARTIN–SCHELTER REGULAR ALGEBRAS

2014 ◽  
Vol 91 (1) ◽  
pp. 53-68 ◽  
Author(s):  
Y. SHEN ◽  
G.-S. ZHOU ◽  
D.-M. LU
Keyword(s):  
Regular Algebras

AbstractWe introduce a method named homogeneous PBW deformation that preserves the regularity and some other homological properties for multigraded algebras. The method is used to produce Artin–Schelter regular algebras without the hypothesis on grading.

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