Pub Date : 2022-02-01DOI: 10.5772/intechopen.101083
Seyed Mohammad Ajdani, F. Bulnes
Monomials are the link between commutative algebra and combinatorics. With a simplicial complex Δ, one can associate two square-free monomial ideals: the Stanley-Reisner ideal IΔ whose generators correspond to the non-face of Δ, or the facet ideal I(Δ) that is a generalization of edge ideals of graphs and whose generators correspond to the facets of Δ. The facet ideal of a simplicial complex was first introduced by Faridi in 2002. Let G be a simple graph. The edge ideal I(G) of a graph G was first considered by R. Villarreal in 1990. He studied algebraic properties of I(G) using a combinatorial language of G. In combinatorial commutative algebra, one can attach a monomial ideal to a combinatorial object. Then, algebraic properties of this ideal are studied using combinatorial properties of combinatorial object. One of interesting problems in combinatorial commutative algebra is the Stanley’s conjectures. The Stanley’s conjectures are studied by many researchers. Let R be a Nn-graded ring and M a Zn-graded R-module. Then, Stanley conjectured that depthM≤sdepthM. He also conjectured that each Cohen-Macaulay simplicial complex is partition-able. In this chapter, we study the relation between vertex decomposability of some simplicial complexes and Stanley’s conjectures.
{"title":"Vertex Decomposability of Path Complexes and Stanley’s Conjectures","authors":"Seyed Mohammad Ajdani, F. Bulnes","doi":"10.5772/intechopen.101083","DOIUrl":"https://doi.org/10.5772/intechopen.101083","url":null,"abstract":"Monomials are the link between commutative algebra and combinatorics. With a simplicial complex Δ, one can associate two square-free monomial ideals: the Stanley-Reisner ideal IΔ whose generators correspond to the non-face of Δ, or the facet ideal I(Δ) that is a generalization of edge ideals of graphs and whose generators correspond to the facets of Δ. The facet ideal of a simplicial complex was first introduced by Faridi in 2002. Let G be a simple graph. The edge ideal I(G) of a graph G was first considered by R. Villarreal in 1990. He studied algebraic properties of I(G) using a combinatorial language of G. In combinatorial commutative algebra, one can attach a monomial ideal to a combinatorial object. Then, algebraic properties of this ideal are studied using combinatorial properties of combinatorial object. One of interesting problems in combinatorial commutative algebra is the Stanley’s conjectures. The Stanley’s conjectures are studied by many researchers. Let R be a Nn-graded ring and M a Zn-graded R-module. Then, Stanley conjectured that depthM≤sdepthM. He also conjectured that each Cohen-Macaulay simplicial complex is partition-able. In this chapter, we study the relation between vertex decomposability of some simplicial complexes and Stanley’s conjectures.","PeriodicalId":206412,"journal":{"name":"Advanced Topics of Topology [Working Title]","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130270933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-26DOI: 10.5772/intechopen.101524
Glaisa T. Catalan, Michael P. Baldado Jr, Roberto N. Padua
Let XτI be an ideal topological space. A subset A of X is said to be β-open if A⊆clintclA, and it is said to be βI-open if there is a set O∈τ with the property 1O−A∈I and 2A−clintclO∈I. The set A is called βI-compact if every cover of A by βI-open sets has a finite sub-cover. The set A is said to be cβI-compact, if every cover Oλ:λ∈Λ of A by β-open sets, Λ has a finite subset Λ0 such that A−∪Oλ:λ∈Λ0∈I. The set A is said to be countably βI-compact if every countable cover of A by βI-open sets has a finite sub-cover. An ideal topological space XτI is said to be βI∗-hyperconnected if X−cl∗A∈I for every non-empty βI-open subset A of X. Two subsets A and B of X is said to be βI-separated if clβIA∩B=∅=A∩clβB. Moreover, A is called a βI-connected set if it can’t be written as a union of two βI-separated subsets. An ideal topological space XτI is called βI-connected space if X is βI-connected. In this article, we give some important properties of βI-open sets, βI-compact spaces, cβI-compact spaces, βI∗-hyperconnected spaces, and βI-connected spaces.
设XτI是一个理想拓扑空间。若存在一个集O∈τ，且其性质为O−A∈I，且2A−clintclO∈I，则称X的一个子集A为β-开放。如果集合A由β i -开集构成的每个覆盖都有有限子覆盖，则称集合A为β i -紧。集合A是cβI-紧的，如果A被β-开集覆盖Oλ:λ∈Λ， Λ有一个有限子集Λ0，使得A−∪Oλ:λ∈Λ0∈I。如果A通过β i开集的每一个可数覆盖都有有限子覆盖，则称集合A是可数β i紧的。对于X的每个非空βI-开子集A，如果X−cl∗A∈I，则理想拓扑空间XτI是βI∗-超连通的，如果lβ ia∩B=∅=A∩lβB，则X的两个子集A和B是βI分离的。此外，如果A不能写成两个β i分离子集的并集，则称为β i连通集。如果X是β i连通的，则理想拓扑空间XτI称为β i连通空间。本文给出了βI开集、βI紧空间、cβI紧空间、βI * -超连通空间和βI连通空间的一些重要性质。
{"title":"βI-Compactness, βI*-Hyperconnectedness and βI-Separatedness in Ideal Topological Spaces","authors":"Glaisa T. Catalan, Michael P. Baldado Jr, Roberto N. Padua","doi":"10.5772/intechopen.101524","DOIUrl":"https://doi.org/10.5772/intechopen.101524","url":null,"abstract":"Let XτI be an ideal topological space. A subset A of X is said to be β-open if A⊆clintclA, and it is said to be βI-open if there is a set O∈τ with the property 1O−A∈I and 2A−clintclO∈I. The set A is called βI-compact if every cover of A by βI-open sets has a finite sub-cover. The set A is said to be cβI-compact, if every cover Oλ:λ∈Λ of A by β-open sets, Λ has a finite subset Λ0 such that A−∪Oλ:λ∈Λ0∈I. The set A is said to be countably βI-compact if every countable cover of A by βI-open sets has a finite sub-cover. An ideal topological space XτI is said to be βI∗-hyperconnected if X−cl∗A∈I for every non-empty βI-open subset A of X. Two subsets A and B of X is said to be βI-separated if clβIA∩B=∅=A∩clβB. Moreover, A is called a βI-connected set if it can’t be written as a union of two βI-separated subsets. An ideal topological space XτI is called βI-connected space if X is βI-connected. In this article, we give some important properties of βI-open sets, βI-compact spaces, cβI-compact spaces, βI∗-hyperconnected spaces, and βI-connected spaces.","PeriodicalId":206412,"journal":{"name":"Advanced Topics of Topology [Working Title]","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120983145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-24DOI: 10.5772/intechopen.101427
Sanjay Kumar Singh, Punam Gupta
In this chapter, we give the detailed study about the Clairaut submersion. The fundamental notations are given. Clairaut submersion is one of the most interesting topics in differential geometry. Depending on the condition on distribution of submersion, we have different classes of submersion such as anti-invariant, semi-invariant submersions etc. We describe the geometric properties of Clairaut anti-invariant submersions and Clairaut semi-invariant submersions whose total space is a Kähler, nearly Kähler manifold. We give condition for Clairaut anti-invariant submersion to be a totally geodesic map and also study Clairaut anti-invariant submersions with totally umbilical fibers. We also give the conditions for the semi-invariant submersions to be Clairaut map and also for Clairaut semi-invariant submersion to be a totally geodesic map. We also give some illustrative example of Clairaut anti-invariant and semi-invariant submersion.
{"title":"Clairaut Submersion","authors":"Sanjay Kumar Singh, Punam Gupta","doi":"10.5772/intechopen.101427","DOIUrl":"https://doi.org/10.5772/intechopen.101427","url":null,"abstract":"In this chapter, we give the detailed study about the Clairaut submersion. The fundamental notations are given. Clairaut submersion is one of the most interesting topics in differential geometry. Depending on the condition on distribution of submersion, we have different classes of submersion such as anti-invariant, semi-invariant submersions etc. We describe the geometric properties of Clairaut anti-invariant submersions and Clairaut semi-invariant submersions whose total space is a Kähler, nearly Kähler manifold. We give condition for Clairaut anti-invariant submersion to be a totally geodesic map and also study Clairaut anti-invariant submersions with totally umbilical fibers. We also give the conditions for the semi-invariant submersions to be Clairaut map and also for Clairaut semi-invariant submersion to be a totally geodesic map. We also give some illustrative example of Clairaut anti-invariant and semi-invariant submersion.","PeriodicalId":206412,"journal":{"name":"Advanced Topics of Topology [Working Title]","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129707607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-11DOI: 10.5772/intechopen.100723
Y. Kamiyama
As a mathematical model for cycloalkenes, we consider equilateral polygons whose interior angles are the same except for those of the both ends of the specified edge. We study the configuration space of such polygons. It is known that for some case, the space is homeomorphic to a sphere. The purpose of this chapter is threefold: First, using the h-cobordism theorem, we prove that the above homeomorphism is in fact a diffeomorphism. Second, we study the best possible condition for the space to be a sphere. At present, only a sphere appears as a topological type of the space. Then our third purpose is to show the case when a closed surface of positive genus appears as a topological type.
{"title":"The Topology of the Configuration Space of a Mathematical Model for Cycloalkenes","authors":"Y. Kamiyama","doi":"10.5772/intechopen.100723","DOIUrl":"https://doi.org/10.5772/intechopen.100723","url":null,"abstract":"As a mathematical model for cycloalkenes, we consider equilateral polygons whose interior angles are the same except for those of the both ends of the specified edge. We study the configuration space of such polygons. It is known that for some case, the space is homeomorphic to a sphere. The purpose of this chapter is threefold: First, using the h-cobordism theorem, we prove that the above homeomorphism is in fact a diffeomorphism. Second, we study the best possible condition for the space to be a sphere. At present, only a sphere appears as a topological type of the space. Then our third purpose is to show the case when a closed surface of positive genus appears as a topological type.","PeriodicalId":206412,"journal":{"name":"Advanced Topics of Topology [Working Title]","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125978173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-13DOI: 10.5772/intechopen.99464
P. Rodrigo, S. Maheswari
The concept of neutrosophic continuous function was very first introduced by A.A. Salama et al. The main aim of this paper is to introduce a new concept of Neutrosophic continuous function namely Strongly Neutrosophic gsα* - continuous functions, Perfectly Neutrosophic gsα* - continuous functions and Totally Neutrosophic gsα* - continuous functions in Neutrosophic topological spaces. These concepts are derived from strongly generalized neutrosophic continuous function and perfectly generalized neutrosophic continuous function. Several interesting properties and characterizations are derived and compared with already existing neutrosophic functions.
{"title":"More Functions Associated with Neutrosophic gsα*- Closed Sets in Neutrosophic Topological Spaces","authors":"P. Rodrigo, S. Maheswari","doi":"10.5772/intechopen.99464","DOIUrl":"https://doi.org/10.5772/intechopen.99464","url":null,"abstract":"The concept of neutrosophic continuous function was very first introduced by A.A. Salama et al. The main aim of this paper is to introduce a new concept of Neutrosophic continuous function namely Strongly Neutrosophic gsα* - continuous functions, Perfectly Neutrosophic gsα* - continuous functions and Totally Neutrosophic gsα* - continuous functions in Neutrosophic topological spaces. These concepts are derived from strongly generalized neutrosophic continuous function and perfectly generalized neutrosophic continuous function. Several interesting properties and characterizations are derived and compared with already existing neutrosophic functions.","PeriodicalId":206412,"journal":{"name":"Advanced Topics of Topology [Working Title]","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125493460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：