Intervals in subgroup lattices of infinite groups

Tůma, Jiří

doi:10.1016/0021-8693(89)90171-3

Cited by 14 publications

(15 citation statements)

References 5 publications

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“…This remains open -for the latter two lattices this question has been posed as an open problem in [GP08] (Problems B and A, respectively). By a deep theorem due to Tůma [Tům89], every complete algebraic lattice with λ compact elements is isomorphic to an interval of the subgroup lattice of a group of size λ; from this it only follows that Gr(λ) contains all complete algebraic lattices with at most λ compact elements as intervals. Proving that Gr(λ) contains all complete algebraic lattices with at most 2 λ compact elements as intervals would be a common strengthening of Tůma's result and a positive answer to Problem 2.1.…”

Section: Related Work and Possible Extensionsmentioning

confidence: 99%

Universality of the lattice of transformation monoids

Pinsker¹,

Shelah

2013

Proc. Amer. Math. Soc.

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Abstract. The set of all transformation monoids on a fixed set of infinite cardinality λ, equipped with the order of inclusion, forms a complete algebraic lattice Mon(λ) with 2 λ compact elements. We show that this lattice is universal with respect to closed sublattices, i.e., the closed sublattices of Mon(λ) are, up to isomorphism, precisely the complete algebraic lattices with at most 2 λ compact elements.

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Section: Related Work and Possible Extensionsmentioning

confidence: 99%

Universality of the lattice of transformation monoids

Pinsker¹,

Shelah

2013

Proc. Amer. Math. Soc.

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“…It has also been shown by J. Tůma [12] that any algebraic lattice with countably many compact elements is an interval in the subgroup lattice of a countable group. This was extended to a locally-finite countable group by V. Repnitskii and J. Tůma [7].…”

Section: History Of Congruence Latticesmentioning

confidence: 99%

“…It is known [12] that any algebraic lattice is an interval in a subgroup lattice, so it is congruences for an algebra of G-sets. Every algebraic lattice therefore admits some graphical algebra that can be congruences of a G-set.…”

Section: Does Every Finite Lattice Admit a Finitary Graphical Algebra?mentioning

confidence: 99%

Graphical algebras — a new approach to congruence lattices

Kenney

2010

Algebra Univers.

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In 1970, H. Werner considered the question of which sublattices of partition lattices are congruence lattices for an algebra on the underlying set of the partition lattices. He showed that a complete sublattice of a partition lattice is a congruence lattice if and only if it is closed under a new operation called graphical composition. We study the properties of this new operation, viewed as an operation on an abstract lattice. We obtain some necessary properties, and we also obtain some sufficient conditions for an operation on an abstract lattice L to be this operation on a congruence lattice isomorphic to L. We use this result to give a new proof of Grätzer and Schmidt's result that any algebraic lattice occurs as a congruence lattice.

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“…[20]. By a much stronger result due to Lampe (see [22]), for any two nontrivial algebraic lattices L, K and any group G there is an algebra A whose subalgebra and congruence lattice is isomorphic to L and K, respectively, and whose automorphism group is isomorphic to G. Moreover, Tuma [32] has shown that every algebraic lattice is isomorphic to an interval of a subgroup lattice Sub(G).…”

Section: Introductionmentioning

confidence: 99%

Characteristic triangles of closure operators with applications in general algebra

et al. 2009

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Abstract. Our aim is to investigate groups and their weak congruence lattices in the abstract setting of lattices L with (local) closure operators C in the categorical sense, where L is regarded as a small category and C is a family of closure maps on the principal ideals of L. A useful tool for structural investigations of such "lattices with closure" is the socalled characteristic triangle, a certain substructure of the square L 2 . For example, a purely order-theoretical investigation of the characteristic triangle shows that the Dedekind groups (alias Hamiltonian groups) are precisely those with modular weak congruence lattices, and similar results are obtained for other classes of algebras.

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Intervals in subgroup lattices of infinite groups

Cited by 14 publications

References 5 publications

Universality of the lattice of transformation monoids

Universality of the lattice of transformation monoids

Graphical algebras — a new approach to congruence lattices

Characteristic triangles of closure operators with applications in general algebra

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