We give here the exact maximal subgroup growthof two classes of polycyclic groups. LetGk=⟨x1, x2, . . . , xk|xixjx−1ixjfor alli < j⟩, soGk=Z ⋊(Z ⋊(Z ⋊· · ·⋊ Z)). Then forall integersk⩾2, we calculatemn(Gk), the number of maximalsubgroups ofGkof indexn, exactly. Also, for inőnitely many groupsHkof the form Z2⋊G2, we calculatemn(Hk)exactly.
{"title":"Maximal subgroup growth of a few polycyclic groups","authors":"A. J. Kelley, Elizabeth Ciorsdan Dwyer Wolfe","doi":"10.12958/adm1506","DOIUrl":"https://doi.org/10.12958/adm1506","url":null,"abstract":"We give here the exact maximal subgroup growthof two classes of polycyclic groups. LetGk=⟨x1, x2, . . . , xk|xixjx−1ixjfor alli < j⟩, soGk=Z ⋊(Z ⋊(Z ⋊· · ·⋊ Z)). Then forall integersk⩾2, we calculatemn(Gk), the number of maximalsubgroups ofGkof indexn, exactly. Also, for inőnitely many groupsHkof the form Z2⋊G2, we calculatemn(Hk)exactly.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44295965","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 obtain an explicit crystal isomorphism between two realizations of crystal bases of finite dimensional irreducible representations of simple Lie algebras of type A and D. The first realization we consider is a geometric construction in terms of irreducible components of certain Nakajima quiver varieties established by Saito and the second is a realization in terms of isomorphism classes of quiver representations obtained by Reineke. We give a homological description of the irreducible components of Lusztig's quiver varieties which correspond to the crystal of a finite dimensional representation and describe the promotion operator in type A to obtain a geometric realization of Kirillov-Reshetikhin crystals.
{"title":"Nakajima quiver varieties, affine crystals and combinatorics of Auslander-Reiten quivers","authors":"Deniz Kus, Bea Schumann","doi":"10.12958/adm1952","DOIUrl":"https://doi.org/10.12958/adm1952","url":null,"abstract":"We obtain an explicit crystal isomorphism between two realizations of crystal bases of finite dimensional irreducible representations of simple Lie algebras of type A and D. The first realization we consider is a geometric construction in terms of irreducible components of certain Nakajima quiver varieties established by Saito and the second is a realization in terms of isomorphism classes of quiver representations obtained by Reineke. We give a homological description of the irreducible components of Lusztig's quiver varieties which correspond to the crystal of a finite dimensional representation and describe the promotion operator in type A to obtain a geometric realization of Kirillov-Reshetikhin crystals.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43160127","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}
Grigorchuk's Overgroup G˜, is a branch group of intermediate growth. It contains the first Grigorchuk's torsion group G of intermediate growth constructed in 1980, but also has elements of infinite order. Its growth is substantially greater than the growth of G. The group G, corresponding to the sequence (012)∞=012012…, is a member of the family {Gω|ω∈Ω={0,1,2}N} consisting of groups of intermediate growth when sequence ω is not eventually constant. Following this construction we define the family {G˜ω,ω∈Ω} of generalized overgroups. Then G˜=G˜(012)∞ and Gω is a subgroup of G˜ω for each ω∈Ω. We prove, if ω is eventually constant, then G˜ω is of polynomial growth and if ω is not eventually constant, then G˜ω is of intermediate growth.
{"title":"On growth of generalized Grigorchuk's overgroups","authors":"Supun T. Samarakoon","doi":"10.12958/adm1451","DOIUrl":"https://doi.org/10.12958/adm1451","url":null,"abstract":"Grigorchuk's Overgroup G˜, is a branch group of intermediate growth. It contains the first Grigorchuk's torsion group G of intermediate growth constructed in 1980, but also has elements of infinite order. Its growth is substantially greater than the growth of G. The group G, corresponding to the sequence (012)∞=012012…, is a member of the family {Gω|ω∈Ω={0,1,2}N} consisting of groups of intermediate growth when sequence ω is not eventually constant. Following this construction we define the family {G˜ω,ω∈Ω} of generalized overgroups. Then G˜=G˜(012)∞ and Gω is a subgroup of G˜ω for each ω∈Ω. We prove, if ω is eventually constant, then G˜ω is of polynomial growth and if ω is not eventually constant, then G˜ω is of intermediate growth.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47682079","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}
A Hausdorff topology τ on the bicyclic monoid with adjoined zero C0 is called weak if it is contained in the coarsest inverse semigroup topology on C0. We show that the lattice W of all weak shift-continuous topologies on C0 is isomorphic to the lattice SIF1×SIF1 where SIF1 is the set of all shift-invariant filters on ω with an attached element 1 endowed with the following partial order: F≤G if and only if G=1 or F⊂G. Also, we investigate cardinal characteristics of the lattice W. In particular, we prove that W contains an antichain of cardinality 2c and a well-ordered chain of cardinality c. Moreover, there exists a well-ordered chain of first-countable weak topologies of order type t.
{"title":"On the lattice of weak topologies on the bicyclic monoid with adjoined zero","authors":"S. Bardyla, O. Gutik","doi":"10.12958/adm1459","DOIUrl":"https://doi.org/10.12958/adm1459","url":null,"abstract":"A Hausdorff topology τ on the bicyclic monoid with adjoined zero C0 is called weak if it is contained in the coarsest inverse semigroup topology on C0. We show that the lattice W of all weak shift-continuous topologies on C0 is isomorphic to the lattice SIF1×SIF1 where SIF1 is the set of all shift-invariant filters on ω with an attached element 1 endowed with the following partial order: F≤G if and only if G=1 or F⊂G. Also, we investigate cardinal characteristics of the lattice W. In particular, we prove that W contains an antichain of cardinality 2c and a well-ordered chain of cardinality c. Moreover, there exists a well-ordered chain of first-countable weak topologies of order type t.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44929656","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 X be a set, BX denotes the family of all subsets of X and F:X→BX be a set-valued mapping such that x∈F(x), supx∈X|F(x)|<κ, supx∈X|F−1(x)|<κ for all x∈X and some infinite cardinal κ. Then there exists a family F of bijective selectors of F such that |F|<κ and F(x)={f(x):f∈F} for each x∈X. We apply this result to G-space representations of balleans.
{"title":"Decompositions of set-valued mappings","authors":"I. Protasov","doi":"10.12958/ADM1485","DOIUrl":"https://doi.org/10.12958/ADM1485","url":null,"abstract":"Let X be a set, BX denotes the family of all subsets of X and F:X→BX be a set-valued mapping such that x∈F(x), supx∈X|F(x)|<κ, supx∈X|F−1(x)|<κ for all x∈X and some infinite cardinal κ. Then there exists a family F of bijective selectors of F such that |F|<κ and F(x)={f(x):f∈F} for each x∈X. We apply this result to G-space representations of balleans.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49269745","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}
Binary linear codes are constructed from graphs, in particular, by the generator matrix [In|A] where A is the adjacency matrix of a graph on n vertices. A combinatorial interpretation of the minimum distance of such codes is given. We also present graph theoretic conditions for such linear codes to be Type I and Type II self-dual. Several examples of binary linear codes produced by well-known graph classes are given.
{"title":"Isodual and self-dual codes from graphs","authors":"Sudipta Mallik, B. Yildiz","doi":"10.12958/adm1645","DOIUrl":"https://doi.org/10.12958/adm1645","url":null,"abstract":"Binary linear codes are constructed from graphs, in particular, by the generator matrix [In|A] where A is the adjacency matrix of a graph on n vertices. A combinatorial interpretation of the minimum distance of such codes is given. We also present graph theoretic conditions for such linear codes to be Type I and Type II self-dual. Several examples of binary linear codes produced by well-known graph classes are given.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49577358","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 (F) be a field, (A) be a vector space over (F) and (G) be a subgroup of (mathrm{GL}(F,A)). We say that (G) has a dense family of subgroups, having finite central dimension, if for every pair of subgroups (H), (K) of (G) such that (Hleqslant K) and (H) is not maximal in (K) there exists a subgroup (L) of finite central dimension such that (Hleqslant Lleqslant K). In this paper we study some locally soluble linear groups with a dense family of subgroups, having finite central dimension.
{"title":"Linear groups saturated by subgroups of finite central dimension","authors":"M. N. Semko, L. Skaskiv, O. A. Yarovaya","doi":"10.12958/ADM1317","DOIUrl":"https://doi.org/10.12958/ADM1317","url":null,"abstract":"Let (F) be a field, (A) be a vector space over (F) and (G) be a subgroup of (mathrm{GL}(F,A)). We say that (G) has a dense family of subgroups, having finite central dimension, if for every pair of subgroups (H), (K) of (G) such that (Hleqslant K) and (H) is not maximal in (K) there exists a subgroup (L) of finite central dimension such that (Hleqslant Lleqslant K). In this paper we study some locally soluble linear groups with a dense family of subgroups, having finite central dimension.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46534219","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 generalize Kudryavtseva and Mazorchuk's concept of a canonical form of elements [9] in Kiselman's semigroups to the setting of a Hecke-Kiselman monoid HKΓ associated with a simple oriented graph Γ. We use confluence properties from [7] to associate with each element in HKΓ a normal form; normal forms are not unique, and we show that they can be obtained from each other by a sequence of elementary commutations. We finally describe a general procedure to recover a (unique) lexicographically minimal normal form.
{"title":"Normal form in Hecke-Kiselman monoids associated with simple oriented graphs","authors":"R. Aragona, Alessandro D'Andrea","doi":"10.12958/adm1571","DOIUrl":"https://doi.org/10.12958/adm1571","url":null,"abstract":"We generalize Kudryavtseva and Mazorchuk's concept of a canonical form of elements [9] in Kiselman's semigroups to the setting of a Hecke-Kiselman monoid HKΓ associated with a simple oriented graph Γ. We use confluence properties from [7] to associate with each element in HKΓ a normal form; normal forms are not unique, and we show that they can be obtained from each other by a sequence of elementary commutations. We finally describe a general procedure to recover a (unique) lexicographically minimal normal form.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45978010","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}
A subset D of an abelian group is decomposable if ∅≠D⊂D+D. In the paper we give partial answers to an open problem asking whether every finite decomposable subset D of an abelian group contains a non-empty subset Z⊂D with ∑Z=0. For every n∈N we present a decomposable subset D of cardinality |D|=n in the cyclic group of order 2n−1 such that ∑D=0, but ∑T≠0 for any proper non-empty subset T⊂D. On the other hand, we prove that every decomposable subset D⊂R of cardinality |D|≤7 contains a non-empty subset T⊂D of cardinality |Z|≤12|D| with ∑Z=0. For every n∈N we present a subset D⊂Z of cardinality |D|=2n such that ∑Z=0 for some subset Z⊂D of cardinality |Z|=n and ∑T≠0 for any non-empty subset T⊂D of cardinality |T|
{"title":"Zero-sum subsets of decomposable sets in Abelian groups","authors":"T. Banakh, A. Ravsky","doi":"10.12958/adm1494","DOIUrl":"https://doi.org/10.12958/adm1494","url":null,"abstract":"A subset D of an abelian group is decomposable if ∅≠D⊂D+D. In the paper we give partial answers to an open problem asking whether every finite decomposable subset D of an abelian group contains a non-empty subset Z⊂D with ∑Z=0. For every n∈N we present a decomposable subset D of cardinality |D|=n in the cyclic group of order 2n−1 such that ∑D=0, but ∑T≠0 for any proper non-empty subset T⊂D. On the other hand, we prove that every decomposable subset D⊂R of cardinality |D|≤7 contains a non-empty subset T⊂D of cardinality |Z|≤12|D| with ∑Z=0. For every n∈N we present a subset D⊂Z of cardinality |D|=2n such that ∑Z=0 for some subset Z⊂D of cardinality |Z|=n and ∑T≠0 for any non-empty subset T⊂D of cardinality |T|<n=12|D|. Also we prove that every finite decomposable subset D of an Abelian group contains two non-empty subsets A,B such that ∑A+∑B=0.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42462990","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 introduce the notion of the automatic logarithm LogA(B) of a finite initial Mealy automaton B, with another automaton A as the base. It allows one to find for any input word w a power n such that B(w)=An(w). The purpose is to study the expanding properties of graphs describing the action of the group generated by A and B on input words of a fixed length interpreted as levels of a regular d-ary rooted tree T. Formally, the automatic logarithm is a single map LogA(B):∂T→Zd from the boundary of the tree to the d-adic integers. Under the assumption that theaction of the automaton A on the tree T is level-transitive andof bounded activity, we show that LogA(B) can be computed bya Moore machine. The distribution of values of the automatic logarithm yields a probabilistic measure μ on ∂T, which in some cases can be computed by a Mealy-type machine (we then say that μ is finite-state). We provide a criterion to determine whether μ is finite-state. A number of examples with A being the adding machine are considered.
{"title":"Automatic logarithm and associated measures","authors":"R. Grigorchuk, R. Kogan, Yaroslav Vorobets","doi":"10.12958/adm2014","DOIUrl":"https://doi.org/10.12958/adm2014","url":null,"abstract":"We introduce the notion of the automatic logarithm LogA(B) of a finite initial Mealy automaton B, with another automaton A as the base. It allows one to find for any input word w a power n such that B(w)=An(w). The purpose is to study the expanding properties of graphs describing the action of the group generated by A and B on input words of a fixed length interpreted as levels of a regular d-ary rooted tree T. Formally, the automatic logarithm is a single map LogA(B):∂T→Zd from the boundary of the tree to the d-adic integers. Under the assumption that theaction of the automaton A on the tree T is level-transitive andof bounded activity, we show that LogA(B) can be computed bya Moore machine. The distribution of values of the automatic logarithm yields a probabilistic measure μ on ∂T, which in some cases can be computed by a Mealy-type machine (we then say that μ is finite-state). We provide a criterion to determine whether μ is finite-state. A number of examples with A being the adding machine are considered.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2018-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48353600","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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