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1.
Weak congruence lattices and semidistributive congruence lattices are both recent topics in universal algebra. This motivates
the main result of the present paper, which asserts that a finite group G is a Dedekind group if and only if the diagonal relation is a join-semidistributive element in the lattice of weak congruences
of G. A variant in terms of subgroups rather than weak congruences is also given. It is pointed out that no similar result is
valid for rings. An open problem and some results on the join-semidistributivity of weak congruence lattices are also included.
This research of the second and third authors was partially supported by Serbian Ministry of Science and Environment, Grant
No. 144011 and by the Provincial Secretariat for Science and Technological Development, Autonomous Province of Vojvodina,
grant ”Lattice methods and applications”. 相似文献
2.
We present diagrammatic schemes characterizing congruence 3-permutable and distributive algebras. We show that a congruence
3-permutable algebra is congruence meetsemidistributive if and only if it is distributive. We characterize varieties of algebras
satisfying the so-called triangular scheme by means of a Maltsev-type condition. 相似文献
3.
A certain class of atomic, semimodular, semisimple partition lattices is studied. It is shown that this class is precisely
the class of congruence lattices of equivalence algebras.
The first author is granted by project POCTI-ISFL-1-143 of the “Centro de álgebra da Universidade de Lisboa”, supported by
FCT and FEDER. 相似文献
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William DeMeo 《Algebra Universalis》2014,72(3):295-298
We present pairs of isotopic algebras with congruence lattices of different sizes, thus answering negatively the question of whether isotopic algebras must have isomorphic congruence lattices. 相似文献
7.
We characterize the congruence permutable MS-algebras in a result analogous to that of Nachbin for distributive lattices.
In our main result we prove, among other things, that the congruence permutability of an MS-algebra L is equivalent to the existence of two binary operations in L which satisfy certain equational properties. We show that the class of all congruence permutable MS-algebras with these additional
operations forms an equational class. Finally we give a Mal’cev term for this new variety.
Presented by J. Berman.
Received May 12, 2005; accepted in final form November 5, 2005.
Research supported by ANPCyT (Contrato Préstamo BID 1201/OC-AR) and SECyT (UNC) 相似文献
8.
Gábor Czédli 《Algebra Universalis》2014,72(3):225-230
Recently, G. Grätzer has raised an interesting problem: Which distributive lattices are congruence lattices of slim semimodular lattices? We give an eight element slim distributive lattice that cannot be represented as the congruence lattice of a slim semimodular lattice. Our lattice demonstrates the difficulty of the problem. 相似文献
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The main result of this paper is a generalization of the classical equivalence between the category of continuous posets and the category of completely distributive lattices, based on the fact that the continuous posets are precisely the spectra of completely distributive lattices. Here we show that for so-called hereditary and union complete subset selections Z, the category of Z-continuous posets is equivalent (via a suitable spectrum functor) to the category of Z-supercompactly generated lattices; these are completely distributive lattices with a join-dense subset of certain Z-hypercompact elements. By appropriate change of the morphisms, these equivalences turn into dualities. We present two different approaches: the first one directly uses the Z-join ideal completion and the Z-below relation; the other combines two known equivalence theorems, namely a topological representation of Z-continuous posets and a general lattice theoretical representation of closure spaces. 相似文献
11.
The paper studies the first homology of finite regular branched coverings of a universal Borromean orbifold called B
4,4,4ℍ3. We investigate the irreducible components of the first homology as a representation space of the finite covering transformation
group G. This gives information on the first betti number of finite coverings of general 3-manifolds by the universality of B
4,4,4. The main result of the paper is a criterion in terms of the irreducible character whether a given irreducible representation
of G is an irreducible component of the first homology when G admits certain symmetries. As a special case of the motivating argument the criterion is applied to principal congruence
subgroups of B
4,4,4. The group theoretic computation shows that most of the, possibly nonprincipal, congruence subgroups are of positive first
Betti number.
This work is partially supported by the Sonderforschungsbereich 288. 相似文献
12.
Jürgen Reinhold 《Applied Categorical Structures》2000,8(1-2):367-376
We discuss the question whether every finite interval in the lattice of all topologies on some set is isomorphic to an interval in the lattice of all topologies on a finite set – or, equivalently, whether the finite intervals in lattices of topologies are, up to isomorphism, exactly the duals of finite intervals in lattices of quasiorders. The answer to this question is in the affirmative at least for finite atomistic lattices. Applying recent results about intervals in lattices of quasiorders, we see that, for example, the five-element modular but non-distributive lattice cannot be an interval in the lattice of topologies. We show that a finite lattice whose greatest element is the join of two atoms is an interval of T
0-topologies iff it is the four-element Boolean lattice or the five-element non-modular lattice. But only the first of these two selfdual lattices is an interval of orders because order intervals are known to be dually locally distributive. 相似文献
13.
Let S be an eventually regular semigroup. The extensively P-partial congruence pairs and P-partial congruence pairs for S are defined. Furthermore, the relationships between the lattice of congruences on S, the lattice of P-partial kernel normal systems for S, the lattice of extensively P-partial kernel normal systems for S and the poset of P-partial congruence pairs for S are explored. 相似文献
14.
We suggest a method for selecting an L-simplex in an L-polyhedron of an n-lattice in Euclidean space. By taking into account the specific form of the condition that a simplex in the lattice is an L-simplex and by considering a simplex selected from an L-polyhedron, we present a new method for describing all types of L-polyhedra in lattices of given dimension n. We apply the method to deduce all types of L-polyhedra in n-dimensional lattices for n=2,3,4, which are already known from previous results. 相似文献
15.
Toby Kenney 《Algebra Universalis》2010,64(3-4):313-338
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. 相似文献
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The set \({{\mathrm{Quo}}}(\mathbf {A})\) of compatible quasiorders (reflexive and transitive relations) of an algebra \(\mathbf {A}\) forms a lattice under inclusion, and the lattice \({{\mathrm{Con}}}(\mathbf {A})\) of congruences of \(\mathbf {A}\) is a sublattice of \({{\mathrm{Quo}}}(\mathbf {A})\). We study how the shape of congruence lattices of algebras in a variety determine the shape of quasiorder lattices in the variety. In particular, we prove that a locally finite variety is congruence distributive [modular] if and only if it is quasiorder distributive [modular]. We show that the same property does not hold for meet semi-distributivity. From tame congruence theory we know that locally finite congruence meet semi-distributive varieties are characterized by having no sublattice of congruence lattices isomorphic to the lattice \(\mathbf {M}_3\). We prove that the same holds for quasiorder lattices of finite algebras in arbitrary congruence meet semi-distributive varieties, but does not hold for quasiorder lattices of infinite algebras even in the variety of semilattices. 相似文献
18.
Norbert Newrly 《Algebra Universalis》1993,30(2):217-220
The problem of characterizing the lattices of equational theories is still unsolved. In this paper it is shown that these lattices are congruence lattices of an algebra whose fundamental operations consist of one monoid operation with right zero and one unary operation.Presented by R. McKenzie. 相似文献
19.
Harry Lakser 《Algebra Universalis》2001,46(4):515-529
An algebra A is said to be a congruence-preserving extension of a subalgebra B if the mapping from the congruence lattice of B to that of A, assigning to each congruence relation β on B the minimal congruence relation on A containing β, is an isomorphism. We give a necessary and sufficient condition on the congruence lattice of a subdirect product
B of finitely many algebras in a congruence-distributive variety that the full direct product be a congruence-preserving extension
of B. We give several applications to congruence lattices of lattices.
Received May 25, 2000; accepted in final form January 22, 2001. 相似文献
20.
Juhani Nieminen 《manuscripta mathematica》1975,15(3):251-259
A set of results analogous to the corresponding results in lattices is derived for neutral ideals in join-semilattices, for congruence relations generated by neutral ideals in join-semilattices, and for permutable congruence relations generated by ideals in modular join-semilattices. 相似文献