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Canadian Journal of Mathematics, 1959
1.1 This paper gives a lattice theoretic investigation of “finiteness“ and “continuity of the lattice operations” in a complemented modular lattice. Although we usually assume that the lattice is-complete for some infinite,3we do not require completeness and continuity, as von Neumann does in his classical memoir on continuous geometry (3); nor do we ...
Amemiya, Ichiro, Halperin, Israel
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1.1 This paper gives a lattice theoretic investigation of “finiteness“ and “continuity of the lattice operations” in a complemented modular lattice. Although we usually assume that the lattice is-complete for some infinite,3we do not require completeness and continuity, as von Neumann does in his classical memoir on continuous geometry (3); nor do we ...
Amemiya, Ichiro, Halperin, Israel
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Modular C11 lattices and lattice preradicals
Journal of Algebra and Its Applications, 2017This paper deals with properties of modular [Formula: see text] lattices involving hereditary preradicals on hereditary classes of modular lattices. Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories.
Albu, Toma, Iosif, Mihai
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DIMENSION MODULES AND MODULAR LATTICES
Journal of Algebra and Its Applications, 2012A module M is called a dimension module if the Goldie (uniform) dimension satisfies the formula u(A + B) + u(A ∩ B) = u(A) + u(B) for arbitrary submodules A, B of M. Dimension modules and related notions were studied by several authors. In this paper, we study them in a more general context of modular lattices with 0 to which the notion of dimension ...
Domagalska, P., Puczyłowski, E. R.
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Congruence lattices of modular lattices
Publicationes Mathematicae Debrecen, 1993It is a well-known result of R. P. Dilworth and of G. Grätzer and the author that every finite distributive lattice is the congruence lattice of some finite lattice. In addition, finite modular lattices have Boolean congruence lattices. In the paper under review the author gives a new proof of the following theorem: Every finite distributive lattice is
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Algebra Universalis, 1990
The algebras \({\mathcal A}\) for which the subalgebra lattice Sub \({\mathcal A}\times {\mathcal A}\) is a modular lattice are studied. It is proved that if \({\mathcal A}\) is supposed to be, in addition, an idempotent algebra, then it is the trivial algebra with the exception of one two-element algebra. Another theorem states that if Sub \({\mathcal
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The algebras \({\mathcal A}\) for which the subalgebra lattice Sub \({\mathcal A}\times {\mathcal A}\) is a modular lattice are studied. It is proved that if \({\mathcal A}\) is supposed to be, in addition, an idempotent algebra, then it is the trivial algebra with the exception of one two-element algebra. Another theorem states that if Sub \({\mathcal
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Homogeneous Modular Lattices are Distributive
Order, 2015A structure \(A\) is homogeneous if any partial isomorphism between finitely generated substructures can be extended to an automorphism of the structure \(A\). \textit{A. Abogatma} and \textit{J. K. Truss} [Order 32, No. 2, 239--243 (2015; Zbl 1348.06004)] have constructed uncountably many contable homogeneous lattices.
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