Let[n] ={1,2, . . . , n} be a finite chain and let Pn (resp.,Tn) be the semigroup of partial transformations on[n] (resp., full transformations on[n]). Let CPn={α∈ Pn: (for allx, y ∈ Dom α)|xα−yα|⩽|x−y|}(resp., CTn={α∈ Tn: (for allx, y∈[n])|xα−yα|⩽|x−y|}) be the subsemigroup of partial contractionmappings on[n](resp., subsemigroup of full contraction mappingson[n]). We characterize all the starred Green’s relations on C Pn and it subsemigroup of order preserving and/or order reversingand subsemigroup of order preserving partial contractions on[n], respectively. We show that the semigroups CPn and CTn, and some of their subsemigroups are left abundant semigroups for all n but not right abundant forn⩾4. We further show that the set ofregular elements of the semigroup CTn and its subsemigroup of order preserving or order reversing full contractions on[n], each formsa regular subsemigroup and an orthodox semigroup, respectively.
{"title":"On certain semigroups of contraction mappings of a finite chain","authors":"A. Umar, M. M. Zubairu","doi":"10.12958/adm1816","DOIUrl":"https://doi.org/10.12958/adm1816","url":null,"abstract":"Let[n] ={1,2, . . . , n} be a finite chain and let Pn (resp.,Tn) be the semigroup of partial transformations on[n] (resp., full transformations on[n]). Let CPn={α∈ Pn: (for allx, y ∈ Dom α)|xα−yα|⩽|x−y|}(resp., CTn={α∈ Tn: (for allx, y∈[n])|xα−yα|⩽|x−y|}) be the subsemigroup of partial contractionmappings on[n](resp., subsemigroup of full contraction mappingson[n]). We characterize all the starred Green’s relations on C Pn and it subsemigroup of order preserving and/or order reversingand subsemigroup of order preserving partial contractions on[n], respectively. We show that the semigroups CPn and CTn, and some of their subsemigroups are left abundant semigroups for all n but not right abundant forn⩾4. We further show that the set ofregular elements of the semigroup CTn and its subsemigroup of order preserving or order reversing full contractions on[n], each formsa regular subsemigroup and an orthodox semigroup, respectively.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66420014","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}
For M∈R-Mod and τ a hereditary torsion theory on the category σ[M] we use the concept of prime and semiprime module defined by Raggi et al. to introduce the concept of τ-pure prime radical Nτ(M)=Nτ as the intersection of all τ-pure prime submodules of M. We give necessary and sufficient conditions for the τ-nilpotence of Nτ(M). We prove that if M is a finitely generated R-module, progenerator in σ[M] and χ≠τ is FIS-invariant torsion theory such that M has τ-Krull dimension, then Nτ is τ-nilpotent.
{"title":"On the nilpotence of the prime radical in module categories","authors":"C. Arellano, J. Castro, J. Ríos","doi":"10.12958/adm1634","DOIUrl":"https://doi.org/10.12958/adm1634","url":null,"abstract":"For M∈R-Mod and τ a hereditary torsion theory on the category σ[M] we use the concept of prime and semiprime module defined by Raggi et al. to introduce the concept of τ-pure prime radical Nτ(M)=Nτ as the intersection of all τ-pure prime submodules of M. We give necessary and sufficient conditions for the τ-nilpotence of Nτ(M). We prove that if M is a finitely generated R-module, progenerator in σ[M] and χ≠τ is FIS-invariant torsion theory such that M has τ-Krull dimension, then Nτ is τ-nilpotent.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66418891","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}
We describe the algebra of derivation of finite-dimensional cyclic Leibniz algebra.
描述了有限维循环莱布尼兹代数的求导代数。
{"title":"On the structure of the algebra of derivations of cyclic Leibniz algebras","authors":"L. A. Kurdachenko, M. Semko, V. Yashchuk","doi":"10.12958/adm1898","DOIUrl":"https://doi.org/10.12958/adm1898","url":null,"abstract":"We describe the algebra of derivation of finite-dimensional cyclic Leibniz algebra.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66420121","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}
Let R be a commutative ring with identity and let M be an R-module. The main purpose of this paper is to introduce and study the notion of S-second submodules of an R-module M as a~generalization of second submodules of M.
{"title":"S-second submodules of a module","authors":"Faranak Farshadifar","doi":"10.12958/adm1437","DOIUrl":"https://doi.org/10.12958/adm1437","url":null,"abstract":"Let R be a commutative ring with identity and let M be an R-module. The main purpose of this paper is to introduce and study the notion of S-second submodules of an R-module M as a~generalization of second submodules of M.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66418003","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}
Modular data are commonly studied in mathematics and physics. A modular datum defines a finite-dimensional representation of the modular group SL2(Z). Cuntz (2007) defined isomorphic integral modular data. Here we discuss isomorphic integral and non-integral modular data as well as non-isomorphic but closely related modular data. In this paper, we give some insights into diagonal torsion matrices associated to modular data.
{"title":"Diagonal torsion matrices associated with modular data","authors":"G. Singh","doi":"10.12958/adm1368","DOIUrl":"https://doi.org/10.12958/adm1368","url":null,"abstract":"Modular data are commonly studied in mathematics and physics. A modular datum defines a finite-dimensional representation of the modular group SL2(Z). Cuntz (2007) defined isomorphic integral modular data. Here we discuss isomorphic integral and non-integral modular data as well as non-isomorphic but closely related modular data. In this paper, we give some insights into diagonal torsion matrices associated to modular data.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66417504","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}
In 2005, the authors described all introduced by them P-critical posets (minimal finite posets with the quadratic Tits form not being positive); up to isomorphism, their number is 132 (75 if duality is considered). Later (in 2014) A. Polak and D. Simson offered an alternative way of proving by using computer algebra tools. In doing this, they defined and described the Tits P-critical posets as a special case of the P-critical posets. In this paper we classify all the non-Tits P-critical posets without complex calculations and without using the list of all P-critical ones.
2005年，作者描述了所有由他们引入的p临界偏序集(二次Tits形式不为正的极小有限偏序集);直到同构，它们的数量是132(如果考虑对偶性，则为75)。后来(2014年)A. Polak和D. Simson通过使用计算机代数工具提供了另一种证明方法。在此过程中，他们定义并描述了Tits p -临界偏序集作为p -临界偏序集的一种特殊情况。在本文中，我们对所有的非tits p临界序集进行了分类，没有进行复杂的计算，也没有使用所有p临界序集的列表。
{"title":"On classifying the non-Tits P-critical posets","authors":"V. M. Bondarenko, M. Styopochkina","doi":"10.12958/adm1912","DOIUrl":"https://doi.org/10.12958/adm1912","url":null,"abstract":"In 2005, the authors described all introduced by them P-critical posets (minimal finite posets with the quadratic Tits form not being positive); up to isomorphism, their number is 132 (75 if duality is considered). Later (in 2014) A. Polak and D. Simson offered an alternative way of proving by using computer algebra tools. In doing this, they defined and described the Tits P-critical posets as a special case of the P-critical posets. In this paper we classify all the non-Tits P-critical posets without complex calculations and without using the list of all P-critical ones.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66420178","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}
We consider the unnormalized and normalized complexes of a simplicial or a cosimplicial object coming from the Dold-Kan correspondence for an idempotent complete additive category (kernels and cokernels are not required). The normalized complex is defined as the image of certain idempotent in the unnormalized complex. We prove that this idempotent is homotopic to identity via homotopy which is expressed via faces and degeneracies. Hence, the normalized and unnormalized complex are homotopy isomorphic to each other. We provide explicit formulae for the homotopy.
{"title":"Homotopy equivalence of normalized and unnormalized complexes, revisited","authors":"V. Lyubashenko, A. Matsui","doi":"10.12958/adm1879","DOIUrl":"https://doi.org/10.12958/adm1879","url":null,"abstract":"We consider the unnormalized and normalized complexes of a simplicial or a cosimplicial object coming from the Dold-Kan correspondence for an idempotent complete additive category (kernels and cokernels are not required). The normalized complex is defined as the image of certain idempotent in the unnormalized complex. We prove that this idempotent is homotopic to identity via homotopy which is expressed via faces and degeneracies. Hence, the normalized and unnormalized complex are homotopy isomorphic to each other. We provide explicit formulae for the homotopy.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66420480","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}
J. G. Rodríguez-Nieto, O. Salazar-Díaz, R. Velásquez
The aim of this paper is to propose two possible ways of defining a g-digroup action and a first approximation to representation theory of g-digroups.
本文的目的是提出定义g-群作用的两种可能方法和g-群表示理论的第一逼近。
{"title":"The structure of g-digroup actions and representation theory","authors":"J. G. Rodríguez-Nieto, O. Salazar-Díaz, R. Velásquez","doi":"10.12958/adm1741","DOIUrl":"https://doi.org/10.12958/adm1741","url":null,"abstract":"The aim of this paper is to propose two possible ways of defining a g-digroup action and a first approximation to representation theory of g-digroups.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66419200","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}
Let A and B be two factor von Neumann algebras. In this paper, we proved that a bijective mapping Φ:A→B satisfies Φ(a∘b+ba∗)=Φ(a)∘Φ(b)+Φ(b)Φ(a)∗ (where ∘ is the special Jordan product on A and B, respectively), for all elements a,b∈A, if and only if Φ is a ∗-ring isomorphism. In particular, if the von Neumann algebras A and B are type I factors, then Φ is a unitary isomorphism or a conjugate unitary isomorphism.
{"title":"Mappings preserving sum of products a∘b+ba∗ on factor von Neumann algebras","authors":"J. M. Ferreira, M. Marietto","doi":"10.12958/ADM1482","DOIUrl":"https://doi.org/10.12958/ADM1482","url":null,"abstract":"Let A and B be two factor von Neumann algebras. In this paper, we proved that a bijective mapping Φ:A→B satisfies Φ(a∘b+ba∗)=Φ(a)∘Φ(b)+Φ(b)Φ(a)∗ (where ∘ is the special Jordan product on A and B, respectively), for all elements a,b∈A, if and only if Φ is a ∗-ring isomorphism. In particular, if the von Neumann algebras A and B are type I factors, then Φ is a unitary isomorphism or a conjugate unitary isomorphism.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66417839","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}
We study properties of cancellation ideals of ring extensions. Let R⊆S be a ring extension. A nonzero S-regular ideal I of R is called a (quasi)-cancellation ideal of the ring extension R⊆S if whenever IB=IC for two S-regular (finitely generated) R-submodules B and C of S, then B=C. We show that a finitely generated ideal I is a cancellation ideal of the ring extension R⊆S if and only if I is S-invertible.
{"title":"Cancellation ideals of a ring extension","authors":"S. Tchamna","doi":"10.12958/adm1424","DOIUrl":"https://doi.org/10.12958/adm1424","url":null,"abstract":"We study properties of cancellation ideals of ring extensions. Let R⊆S be a ring extension. A nonzero S-regular ideal I of R is called a (quasi)-cancellation ideal of the ring extension R⊆S if whenever IB=IC for two S-regular (finitely generated) R-submodules B and C of S, then B=C. We show that a finitely generated ideal I is a cancellation ideal of the ring extension R⊆S if and only if I is S-invertible.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66417120","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}
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