{"title":"Graver bases of shifted numerical semigroups with 3 generators","authors":"James Howard, Christopher O’Neill","doi":"10.1142/s0219498825502275","DOIUrl":"https://doi.org/10.1142/s0219498825502275","url":null,"abstract":"<p>A numerical semigroup <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>M</mi></math></span><span></span> is a subset of the non-negative integers that is closed under addition. A factorization of <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>∈</mo><mi>M</mi></math></span><span></span> is an expression of <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span> as a sum of generators of <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>M</mi></math></span><span></span>, and the Graver basis of <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>M</mi></math></span><span></span> is a collection <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">Gr</mtext></mstyle><mo stretchy=\"false\">(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span> of trades between the generators of <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>M</mi></math></span><span></span> that allows for efficient movement between factorizations. Given positive integers <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span><span></span>, consider the family <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mo stretchy=\"false\">〈</mo><mi>t</mi><mo stretchy=\"false\">+</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><mi>t</mi><mo stretchy=\"false\">+</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span><span></span> of “shifted” numerical semigroups whose generators are obtained by translating <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span><span></span> by an integer parameter <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>t</mi></math></span><span></span>. In this paper, we characterize the Graver basis <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">Gr</mtext></mstyle><mo stretchy=\"false\">(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span> of <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span><span></span> for sufficiently large <span><math altimg=\"eq-00014.gif\" display=\"inline\" overflow=\"scroll\"><mi>t</mi></math></span><span></span> in the case ","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"38 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140634619","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Cohomology and the controlling algebra of crossed homomorphisms on 3-Lie algebras","authors":"Shuai Hou, Meiyan Hu, Lina Song, Yanqiu Zhou","doi":"10.1142/s0219498825502317","DOIUrl":"https://doi.org/10.1142/s0219498825502317","url":null,"abstract":"<p>In this paper, first we give the notion of a crossed homomorphism on a <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mn>3</mn></math></span><span></span>-Lie algebra with respect to an action on another <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mn>3</mn></math></span><span></span>-Lie algebra, and characterize it using a homomorphism from a 3-Lie algebra to the semidirect product 3-Lie algebra. We also establish the relationship between crossed homomorphisms and relative Rota–Baxter operators of weight <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn></math></span><span></span> on 3-Lie algebras. Next we construct a cohomology theory for a crossed homomorphism on <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mn>3</mn></math></span><span></span>-Lie algebras and classify infinitesimal deformations of crossed homomorphisms using the second cohomology group. Finally, using the higher derived brackets, we construct an <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub></math></span><span></span>-algebra whose Maurer–Cartan elements are crossed homomorphisms. Consequently, we obtain the twisted <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub></math></span><span></span>-algebra that controls deformations of a given crossed homomorphism on <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mn>3</mn></math></span><span></span>-Lie algebras.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"56 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140608944","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Simple modules over quantized Weyl algebras at roots of unity","authors":"Snehashis Mukherjee, Sanu Bera","doi":"10.1142/s0219498825502238","DOIUrl":"https://doi.org/10.1142/s0219498825502238","url":null,"abstract":"<p>In this paper, the simple modules over the second quantized Weyl algebras at roots of unity over an algebraically closed field are classified.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"59 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140589318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Multi-graded Macaulay dual spaces","authors":"Joseph Cummings, Jonathan D. Hauenstein","doi":"10.1142/s0219498825502469","DOIUrl":"https://doi.org/10.1142/s0219498825502469","url":null,"abstract":"<p>We describe an algorithm for computing Macaulay dual spaces for multi-graded ideals. For homogeneous ideals, the natural grading is inherited by the Macaulay dual space which has been leveraged to develop algorithms to compute the Macaulay dual space in each homogeneous degree. Our main theoretical result extends this idea to multi-graded Macaulay dual spaces inherited from multi-graded ideals. This natural duality allows ideal operations to be translated from homogeneous ideals to their corresponding operations on the multi-graded Macaulay dual spaces. In particular, we describe a linear operator with a right inverse for computing quotients by a multi-graded polynomial. By using a total ordering on the homogeneous components of the Macaulay dual space, we also describe how to recursively construct a basis for each component. Several examples are included to demonstrate this new approach.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"35 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140576399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Ad-invariant metrics on nonnice nilpotent Lie algebras","authors":"D. Conti, V. del Barco, F. A. Rossi","doi":"10.1142/s0219498825502329","DOIUrl":"https://doi.org/10.1142/s0219498825502329","url":null,"abstract":"<p>We proved in previous work that all real nilpotent Lie algebras of dimension up to <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn><mn>0</mn></math></span><span></span> carrying an ad-invariant metric are nice, i.e. they admit a nice basis in the sense of Lauret <i>et al.</i> In this paper, we show by constructing explicit examples that nonnice irreducible nilpotent Lie algebras admitting an ad-invariant metric exist for every dimension greater than <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn><mn>0</mn></math></span><span></span> and every nilpotency step greater than <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mn>2</mn></math></span><span></span>. In the way of doing so, we introduce a method to construct Lie algebras with ad-invariant metrics called the single extension, as a parallel to the well-known double extension procedure.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"18 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140576550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Local and 2-local derivations on filiform associative algebras","authors":"Kobiljon Abdurasulov, Shavkat Ayupov, Bakhtiyor Yusupov","doi":"10.1142/s0219498825502421","DOIUrl":"https://doi.org/10.1142/s0219498825502421","url":null,"abstract":"<p>This paper is devoted to the study of local and 2-local derivations of null-filiform, filiform and naturally graded quasi-filiform associative algebras. We prove that these algebras as a rule admit local derivations which are not derivations. We show that filiform and naturally graded quasi-filiform associative algebras admit 2-local derivations which are not derivations and any 2-local derivation of null-filiform associative algebras is a derivation.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"31 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140576552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Study of a division-like property","authors":"Robin Khanfir, Béranger Seguin","doi":"10.1142/s0219498825502214","DOIUrl":"https://doi.org/10.1142/s0219498825502214","url":null,"abstract":"<p>We study a weak divisibility property for noncommutative rings: A nontrivial ring is <i>fadelian</i> if for all nonzero <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>a</mi></math></span><span></span> and <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>x</mi></math></span><span></span> there exist <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>b</mi><mo>,</mo><mi>c</mi></math></span><span></span> such that <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>x</mi><mo>=</mo><mi>a</mi><mi>b</mi><mo stretchy=\"false\">+</mo><mi>c</mi><mi>a</mi></math></span><span></span>. We prove properties of fadelian rings and construct examples thereof which are not division rings, as well as non-Noetherian and non-Ore examples.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"31 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140576553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Haar measure of a profinite n-ary group","authors":"M. Shahryari, M. Rostami","doi":"10.1142/s0219498825502196","DOIUrl":"https://doi.org/10.1142/s0219498825502196","url":null,"abstract":"<p>We prove that every profinite <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span>-ary group <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><mi>G</mi><mo>,</mo><mi>f</mi><mo stretchy=\"false\">)</mo><mo>=</mo><msub><mrow><mstyle><mtext mathvariant=\"normal\">der</mtext></mstyle></mrow><mrow><mi>𝜃</mi><mo>,</mo><mi>b</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>G</mi><mo>,</mo><mo stretchy=\"false\">•</mo><mo stretchy=\"false\">)</mo></math></span><span></span> has a unique Haar measure <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span><span></span> and further for every measurable subset <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>A</mi><mo>⊆</mo><mi>G</mi></math></span><span></span>, we have <disp-formula-group><span><math altimg=\"eq-00007.gif\" display=\"block\" overflow=\"scroll\"><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>A</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>m</mi><mo stretchy=\"false\">(</mo><mi>A</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">−</mo><mn>1</mn><mo stretchy=\"false\">)</mo><msup><mrow><mi>m</mi></mrow><mrow><mo stretchy=\"false\">∗</mo></mrow></msup><mo stretchy=\"false\">(</mo><mi>A</mi><mo stretchy=\"false\">)</mo><mo>,</mo></mrow></math></span><span></span></disp-formula-group> where <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>m</mi></math></span><span></span> and <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>m</mi></mrow><mrow><mo stretchy=\"false\">∗</mo></mrow></msup></math></span><span></span> are the normalized Haar measures of the profinite groups <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><mi>G</mi><mo>,</mo><mo stretchy=\"false\">•</mo><mo stretchy=\"false\">)</mo></math></span><span></span> and the Post cover <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>G</mi></mrow><mrow><mo stretchy=\"false\">∗</mo></mrow></msup></math></span><span></span>, respectively.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"11 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140297923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On (naturally) semifull and (semi)separable semifunctors","authors":"Lucrezia Bottegoni","doi":"10.1142/s0219498825502111","DOIUrl":"https://doi.org/10.1142/s0219498825502111","url":null,"abstract":"<p>The notion of semifunctor between categories, due to [9], is defined as a functor that does not necessarily preserve identities. In this paper, we study how several properties of functors, such as fullness, full faithfulness, separability, natural fullness, can be formulated for semifunctors. Since a full semifunctor is actually a functor, we are led to introduce a notion of semifullness (and then semifull faithfulness) for semifunctors. In order to show that these conditions can be derived from requirements on the hom-set components associated with a semifunctor, we look at “semisplitting properties” for seminatural tranformations and we investigate the corresponding properties for morphisms whose source or target is the image of a semifunctor. We define the notion of naturally semifull semifunctor and we characterize natural semifullness for semifunctors that are part of a semiadjunction in terms of semisplitting conditions for the unit and counit attached to the semiadjunction. We study the behavior of semifunctors with respect to (semi)separability and we prove Rafael-type Theorems for (semi)separable semifunctors and a Maschke-type theorem for separable semifunctors. We provide examples of semifunctors on which we test the properties considered so far.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"158 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204357","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Cryptanalysis of a key exchange protocol based on a congruence-simple semiring action","authors":"A. Otero Sánchez, J. A. López Ramos","doi":"10.1142/s0219498825502299","DOIUrl":"https://doi.org/10.1142/s0219498825502299","url":null,"abstract":"<p>We show that a previously introduced key exchange based on a congruence-simple semiring action is not secure by providing an attack that reveals the shared key from the distributed public information for any of such semirings.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"8 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204356","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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