Some Lattice-Theoretical Properties of Groups and Semi-Groups

Arshinov, M N; Sadovskii, L E

doi:10.1070/rm1972v027n06abeh001394

Cited by 3 publications

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“…It is common knowledge that the structure of a group G is closely connected with the structure of its subgroup lattice Sub G. The literature concerning investigations of lattices of subgroups is extensive and systematized in a series of books and surveys, such as [1,8,10,11,12]. The main aspects considered there are the following: describing of groups with certain types of subgroup lattices, lattice characteristics, lattice isomorphisms.…”

Section: Preliminaries and Formulation Of Resultsmentioning

confidence: 99%

On the representation of lattices by subgroup lattices

Repnitskiî¹

1997

Algebra univers.

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Section: Preliminaries and Formulation Of Resultsmentioning

confidence: 99%

On the representation of lattices by subgroup lattices

Repnitskiî¹

1997

Algebra univers.

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Hazewinkel¹

1993

Encyclopaedia of Mathematics

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Introduction. In [6 Grothendieck et al.], chapter 16, there is a comprehensive discussion of (higher order left) differential operators on schemes and hence on commutative algebras. Even more details can be found in [5 Gabriel], especially in connection with group schemes and their infinitesimal structure.These discussions include a recursive description of these algebras of differential operators in terms of commutators with (left) multiplication operators; see propositions 16.8.8 and 16.8.9 on pages 42 and 43 of [6 Grothendieck et al.]. This description will be recalled later (sections 6, 7; for the affine case only).More or less recently there has arisen substantial interest in whether something more or less similar can be done for non-commutative algebras. Straightforward generalizations definitely do not quite work.The contexts in which this interest arises are, among more, Batalin-Vilkovisky algebras, Gerstenhaber algebras, homotopy algebras, higher derived algebras. See [1 Akman; 2 Akman; 3 Akman et al.] and, especially, the references in these papers. Also the deformation theory of algebras and diagrams à la Gerstenhaber-Shack, [4 Doubek]. The context 'homotopy algebras' is currently probably the most important. For some of this look at the last sections of [9 Markl].There is in addition interest in such algebras in the realm of formal groups; for that consult the chapter on 'divided power algebras and sequences and algebras of differential operators' in [8 Hazewinkel].The setting for this whole note is that of associative algebras over a base ring k that is commutative and unital. Often it is good to think of k as the ring of integers Z. Example. Algebras of differential operators on k[X,Y ] . Certainly a very natural algebra of differential operators on k[X,Y] is formed by the collection of operators that are finite sums of the formLet me point out a few features of this algebra, which I will denote Diff naive (k[X,Y ]) 1 . First of all it is indeed a unital associative (and non-commutative) algebra over k. Second, there is a notion of degree. An expression like (2.1) is of degree ! n if f i, j = 0 for all i + j > n . And this notion satisfies that if D and ! D are of degree ! n,!!! " n , respectively, then D ! D is of degree ! n + n' . Third, it contains all derivations and obvious higher derivations on k[X,Y ] . Fourth, if D is an element of Diff naive (k[X,Y ]) of degree ! n and t is an element of k[X,Y ] , i.e. a polynomial, then the commutator 1.!The reason for the subscript 'naive' will become clear later.

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1994

Subgroup Lattices of Groups

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Some Lattice-Theoretical Properties of Groups and Semi-Groups

Cited by 3 publications

References 15 publications

On the representation of lattices by subgroup lattices

On the representation of lattices by subgroup lattices

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