{"title":"A note on odd periodic derived Hall algebras","authors":"Haicheng Zhang, Xinran Zhang, Zhiwei Zhu","doi":"10.1142/s0219498825502822","DOIUrl":"https://doi.org/10.1142/s0219498825502822","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00001.gif\" display=\"inline\"><mi>m</mi></math></span><span></span> be an odd positive integer and <span><math altimg=\"eq-00002.gif\" display=\"inline\"><msub><mrow><mi>D</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi mathvariant=\"cal\">𝒜</mi><mo stretchy=\"false\">)</mo></math></span><span></span> be the <span><math altimg=\"eq-00003.gif\" display=\"inline\"><mi>m</mi></math></span><span></span>-periodic derived category of a finitary hereditary Abelian category <span><math altimg=\"eq-00004.gif\" display=\"inline\"><mi mathvariant=\"cal\">𝒜</mi></math></span><span></span>. In this note, we prove that there is an embedding of algebras from the derived Hall algebra of <span><math altimg=\"eq-00005.gif\" display=\"inline\"><msub><mrow><mi>D</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi mathvariant=\"cal\">𝒜</mi><mo stretchy=\"false\">)</mo></math></span><span></span> defined by Xu–Chen [Hall algebras of odd periodic triangulated categories, <i>Algebr. Represent. Theory</i><b>16</b>(3) (2013) 673–687] to the extended derived Hall algebra of <span><math altimg=\"eq-00006.gif\" display=\"inline\"><msub><mrow><mi>D</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi mathvariant=\"cal\">𝒜</mi><mo stretchy=\"false\">)</mo></math></span><span></span> defined in [H. Zhang, Periodic derived Hall algebras of hereditary Abelian categories, preprint (2023), arXiv:2303.02912v2]. This homomorphism is given on basis elements, rather than just on generating elements.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"24 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551123","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":"Abelian groups whose endomorphism rings are V-rings","authors":"Afshin Amini, Babak Amini, Ehsan Momtahan","doi":"10.1142/s0219498825502871","DOIUrl":"https://doi.org/10.1142/s0219498825502871","url":null,"abstract":"<p>We study Abelian groups whose endomorphism rings are V-rings. Let <span><math altimg=\"eq-00001.gif\" display=\"inline\"><mi>G</mi></math></span><span></span> be a non-reduced Abelian group, We prove that <span><math altimg=\"eq-00002.gif\" display=\"inline\"><mstyle><mtext mathvariant=\"normal\">End</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math></span><span></span> is a V-ring on either side if and only if <span><math altimg=\"eq-00003.gif\" display=\"inline\"><mi>G</mi><mo>=</mo><mi>B</mi><mo stretchy=\"false\">⊕</mo><msup><mrow><mi>ℚ</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></span> where <span><math altimg=\"eq-00004.gif\" display=\"inline\"><mi>B</mi></math></span><span></span> is a tame elementary Abelian group. We observe that a reduced group whose endomorphism is a V-ring, is an sp-group. Recognizing that <span><math altimg=\"eq-00005.gif\" display=\"inline\"><mstyle><mtext mathvariant=\"normal\">End</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math></span><span></span> is also an sp-group of <span><math altimg=\"eq-00006.gif\" display=\"inline\"><msub><mrow><mo>∏</mo></mrow><mrow><mi>p</mi><mo>∈</mo><mi>ℙ</mi></mrow></msub><mstyle><mtext mathvariant=\"normal\">End</mtext></mstyle><mo stretchy=\"false\">(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span>, we show that <span><math altimg=\"eq-00007.gif\" display=\"inline\"><mstyle><mtext mathvariant=\"normal\">End</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">/</mo><mo stretchy=\"false\">⊕</mo><mstyle><mtext mathvariant=\"normal\">End</mtext></mstyle><mo stretchy=\"false\">(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span> is a V-ring if and only if <span><math altimg=\"eq-00008.gif\" display=\"inline\"><mstyle><mtext mathvariant=\"normal\">End</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math></span><span></span> is a V-ring.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"46 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551125","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":"Künneth formulas for Cotor","authors":"A. Salch","doi":"10.1142/s0219498825502652","DOIUrl":"https://doi.org/10.1142/s0219498825502652","url":null,"abstract":"<p>We investigate the question of how to compute the cotensor product, and more generally the derived cotensor (i.e. Cotor) groups, of a tensor product of comodules. In particular, we determine the conditions under which there is a Künneth formula for Cotor. We show that there is a simple Künneth theorem for Cotor groups if and only if an appropriate coefficient comodule has trivial coaction. This result is an application of a spectral sequence we construct for computing Cotor of a tensor product of comodules. Finally, for certain families of nontrivial comodules which are especially topologically natural, we work out necessary and sufficient conditions for the existence of a Künneth formula for the <span><math altimg=\"eq-00001.gif\" display=\"inline\"><mn>0</mn></math></span><span></span>th Cotor group, i.e. the cotensor product. We give topological applications in the form of consequences for the <span><math altimg=\"eq-00002.gif\" display=\"inline\"><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span><span></span>-term of the Adams spectral sequence of a smash product of spectra, and the Hurewicz image of a smash product of spectra.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"17 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551124","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":"Near automorphisms of the complement or the square of a cycle","authors":"Jinxing Zhao","doi":"10.1142/s021949882550286x","DOIUrl":"https://doi.org/10.1142/s021949882550286x","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00001.gif\" display=\"inline\"><mi>G</mi></math></span><span></span> be a graph with vertex set <span><math altimg=\"eq-00002.gif\" display=\"inline\"><mi>V</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, <span><math altimg=\"eq-00003.gif\" display=\"inline\"><mi>f</mi></math></span><span></span> a permutation of <span><math altimg=\"eq-00004.gif\" display=\"inline\"><mi>V</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math></span><span></span>. Define <span><math altimg=\"eq-00005.gif\" display=\"inline\"><msub><mrow><mi>δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo>|</mo><mi>d</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">−</mo><mi>d</mi><mo stretchy=\"false\">(</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>,</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>y</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mo>|</mo></math></span><span></span> and <span><math altimg=\"eq-00006.gif\" display=\"inline\"><msub><mrow><mi>δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo>∑</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, where the sum is taken over all unordered pairs <span><math altimg=\"eq-00007.gif\" display=\"inline\"><mi>x</mi></math></span><span></span>, <span><math altimg=\"eq-00008.gif\" display=\"inline\"><mi>y</mi></math></span><span></span> of distinct vertices of <span><math altimg=\"eq-00009.gif\" display=\"inline\"><mi>G</mi></math></span><span></span>. Let <span><math altimg=\"eq-00010.gif\" display=\"inline\"><mi>π</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math></span><span></span> denote the smallest positive value of <span><math altimg=\"eq-00011.gif\" display=\"inline\"><msub><mrow><mi>δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math></span><span></span> among all permutations <span><math altimg=\"eq-00012.gif\" display=\"inline\"><mi>f</mi></math></span><span></span> of <span><math altimg=\"eq-00013.gif\" display=\"inline\"><mi>V</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math></span><span></span>. A permutation <span><math altimg=\"eq-00014.gif\" display=\"inline\"><mi>f</mi></math></span><span></span> with <span><math altimg=\"eq-00015.gif\" display=\"inline\"><msub><mrow><mi>δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>π</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math></span><span></span> is called a near automorphism of <span><math altimg=\"eq-00016.gif\" display=\"inline\"><mi>G<","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"93 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141257003","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":"Semirings generated by idempotents","authors":"David Dolžan","doi":"10.1142/s0219498825502846","DOIUrl":"https://doi.org/10.1142/s0219498825502846","url":null,"abstract":"<p>We prove that a semiring multiplicatively generated by its idempotents is commutative and Boolean, if every idempotent in the semiring has an orthogonal complement. We prove that a semiring additively generated by its idempotents is commutative, if every idempotent in the semiring has an orthogonal complement and all the nilpotents in the semirings are central. We also provide examples that the assumptions on the existence of orthogonal complements of idempotents and the centrality of nilpotents cannot be omitted.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"23 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141257111","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":"Algebraic K0 for unpointed categories","authors":"Felix Küng","doi":"10.1142/s0219498825502743","DOIUrl":"https://doi.org/10.1142/s0219498825502743","url":null,"abstract":"<p>We construct a natural generalization of the Grothendieck group <span><math altimg=\"eq-00003.gif\" display=\"inline\"><msub><mrow><mstyle><mtext mathvariant=\"normal\">K</mtext></mstyle></mrow><mrow><mn>0</mn></mrow></msub></math></span><span></span> to the case of possibly unpointed categories admitting pushouts by using the concept of heaps recently introduced by Brezinzki. In case of a monoidal category, the defined K0 is shown to be a truss. It is shown that the construction generalizes the classical <span><math altimg=\"eq-00004.gif\" display=\"inline\"><msub><mrow><mstyle><mtext mathvariant=\"normal\">K</mtext></mstyle></mrow><mrow><mn>0</mn></mrow></msub></math></span><span></span> of an abelian category as the group retract along the isomorphism class of the zero object. We finish by applying this construction to construct the integers with addition and multiplication as the decategorification of finite sets and show that in this <span><math altimg=\"eq-00005.gif\" display=\"inline\"><msub><mrow><mstyle><mtext mathvariant=\"normal\">K</mtext></mstyle></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy=\"false\">(</mo><munder accentunder=\"false\"><mrow><mstyle><mtext mathvariant=\"normal\">Top</mtext></mstyle></mrow><mo accent=\"true\">̲</mo></munder><mo stretchy=\"false\">)</mo></math></span><span></span> one can identify a CW-complex with the iterated product of its cells.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"19 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141257184","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 upper bounds for asymptotic ideal-grade","authors":"Saeed Jahandoust","doi":"10.1142/s021949882550272x","DOIUrl":"https://doi.org/10.1142/s021949882550272x","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>I</mi></math></span><span></span> and <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>J</mi></math></span><span></span> be ideals in a Noetherian ring <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>R</mi></math></span><span></span> and let <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span><span></span> be nonunits in <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>R</mi></math></span><span></span>. Then <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span><span></span> is said to be an asymptotic sequence over <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mi>I</mi></math></span><span></span> if <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><mi>I</mi><mo>,</mo><mo stretchy=\"false\">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mi>R</mi><mo>≠</mo><mi>R</mi></math></span><span></span> and if for all <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></math></span><span></span>, <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span><span></span> is not in any associated prime of the integral closure <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><mover accent=\"false\"><mrow><msup><mrow><mo stretchy=\"false\">(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi><mo stretchy=\"false\">−</mo><mn>1</mn></mrow></msub><mo stretchy=\"false\">)</mo></mrow><mrow><mi>m</mi></mrow></msup></mrow><mo accent=\"true\">¯</mo></mover></math></span><span></span> of <span><math altimg=\"eq-00014.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mo stretchy=\"false\">(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi><mo stretchy=\"false\">−</mo><mn>1</mn></mrow></msub><mo stretchy=\"false\">)</mo></mrow><mrow><mi>m</mi></mrow></msup><mo>=</mo><msup><mrow><mo stretchy=\"false\">(</mo><mi>I</mi><mo>,</mo><mo stretchy=\"false\">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo stretchy=\"false\">−</mo><mn>1</mn></mrow></msub><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></mrow><mrow><mi>m</mi></mrow></msup><mi>R</mi></math></span><span></span>, where <span><math altimg=\"eq-00015.gi","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"52 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141152406","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}
Andrey R. Chekhlov, Peter V. Danchev, Patrick W. Keef
{"title":"Semi-generalized co-Bassian groups","authors":"Andrey R. Chekhlov, Peter V. Danchev, Patrick W. Keef","doi":"10.1142/s0219498825502809","DOIUrl":"https://doi.org/10.1142/s0219498825502809","url":null,"abstract":"<p>As a common nontrivial generalization of the notion of a generalized co-Bassian group, recently defined by the third author, we introduce the notion of a <i>semi-generalized co-Bassian</i> group and initiate its comprehensive study. Specifically, we give a complete characterization of these groups in the cases of <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span>-torsion groups and groups of finite torsion-free rank by showing that these groups can be completely determined in terms of generalized finite <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span>-ranks and also depends on their quotients modulo the maximal torsion subgroup. Surprisingly, for <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span>-primary groups, the concept of a semi-generalized co-Bassian group is closely related to that of a generalized co-Bassian group.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"6 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141152405","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":"Morita equivalence and globalization for partial Hopf actions on nonunital algebras","authors":"Marcelo Muniz Alves, Tiago Luiz Ferrazza","doi":"10.1142/s0219498825502627","DOIUrl":"https://doi.org/10.1142/s0219498825502627","url":null,"abstract":"<p>In this work, we investigate partial actions of a Hopf algebra <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>H</mi></math></span><span></span> on nonunital algebras and the associated partial smash products, with the objective of providing a framework where one may obtain results for both <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mo>𝕜</mo></math></span><span></span>-algebras with local units and <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mo>𝕜</mo></math></span><span></span>-categories. We show that our partial actions correspond to nonunital algebras in the category of partial representations of <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>H</mi></math></span><span></span>. The central problem of existence of a globalization for a partial action is studied in detail, and we provide sufficient conditions for the existence (and uniqueness) of a minimal globalization for associative algebras in general. Extending previous results by Abadie, Dokuchaev, Exel and Simon, we define Morita equivalence for partial Hopf actions, and we show that if two symmetrical partial actions are Morita equivalent then their standard globalizations are also Morita equivalent. Particularizing to the case of a partial action on an algebra with local units, we obtain several strong results on equivalences of categories of modules of partial smash products of algebras and partial smash products of <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mo>𝕜</mo></math></span><span></span>-categories.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"100 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141152442","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":"Wilf inequality is preserved under gluing of semigroups","authors":"Srishti Singh, Hema Srinivasan","doi":"10.1142/s021949882550255x","DOIUrl":"https://doi.org/10.1142/s021949882550255x","url":null,"abstract":"<p>Wilf Conjecture on numerical semigroups is a question posed by Wilf in 1978 and is an inequality connecting the Frobenius number, embedding dimension and the genus of the semigroup. The conjecture is still open in general. We prove that this Wilf inequality is preserved under gluing of numerical semigroups. If the numerical semigroups minimally generated by <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>A</mi><mo>=</mo><mo stretchy=\"false\">{</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy=\"false\">}</mo></math></span><span></span> and <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>B</mi><mo>=</mo><mo stretchy=\"false\">{</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy=\"false\">}</mo></math></span><span></span> satisfy the Wilf inequality, then so does their gluing which is minimally generated by <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>C</mi><mo>=</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>A</mi><mo stretchy=\"false\">⊔</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>B</mi></math></span><span></span>. We discuss the extended Wilf’s Conjecture in higher dimensions for certain affine semigroups and prove an analogous result.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"37 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140811111","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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