Constructing the lattice of subalgebras of the dijkstra algebra

It is well known (1, p. 162) that the lattice of subalgebras of a finite Boolean algebra is dually isomorphic to a finite partition lattice. In this paper we study the lattice of subalgebras of an arbitrary Boolean algebra. One of our main results is that the lattice of subalgebras characterizes the Boolean algebra. In order to prove this result we introduce some notions which enable us to give a characterization and representation of the lattices of subalgebras of a Boolean algebra in terms of a closure operator on the lattice of partitions of the Boolean space associated with the Boolean algebra. Our theory then has some analogy to that of the lattice theory of topological vector spaces. Of some interest is the problem of classification of Boolean algebras in terms of the properties of their lattice of subalgebras, and we obtain some results in this direction.

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Associative Algebras with a Distributive Lattice of Subalgebras

Algebra and Logic ◽

10.1007/s10469-020-09608-6 ◽

2020 ◽

Vol 59 (5) ◽

pp. 349-356

Author(s):

A. G. Gein

Keyword(s):

Distributive Lattice ◽

Associative Algebras ◽

Lattice Of Subalgebras

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Modularity* in Lie algebras

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500023464 ◽

1997 ◽

Vol 40 (1) ◽

pp. 99-110 ◽

Cited By ~ 4

Author(s):

K. Bowman ◽

V. R. Varea

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Lattice Of Subalgebras

A subalgebra U of a Lie algebra L over a field F is called modular* in L if U satisfies the dual of the modular identities in the lattice of subalgebras of L. Our aim is the study of the influence of the modular* identities in the structure of the algebra. First we prove that if the modular* conditions are imposed on an ideal of L then every element of L acts as an scalar on this ideal and if they are imposed on a non-ideal subalgebra U of L then the largest ideal of L contained in U also satisfies the modular* identities. We determine Lie algebras having a subalgebra which satisfies both the modular and modular* identities, provided that either L is solvable or char(F)≠ 2,3. As immediate consequences of this result we obtain that the existence of a co-atom satisfying the modular* identities in the lattice L(L) forces that the lattice L(L) is modular and that the modular* identities on any subalgebra U forces that U is quasi-abelian. In the case when L is supersolvable we obtain that the modular* conditions on any non-ideal of L are enough to guarantee that L(L) is modular. For arbitrary fields and any Lie algebra L, we prove that the modular* conditions on every co-atom of the lattice L(L) guarantee that L(L) is modular.

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IRREDUCIBLE ELEMENTS IN ALGEBRAIC LATTICES

International Journal of Algebra and Computation ◽

10.1142/s0218196710005984 ◽

2010 ◽

Vol 20 (08) ◽

pp. 969-975 ◽

Cited By ~ 1

Author(s):

U. M. SWAMY ◽

B. VENKATESWARLU

Keyword(s):

Universal Algebra ◽

Complete Lattice ◽

Sufficient Conditions ◽

Algebraic Lattice ◽

Complete Distributivity ◽

Lattice Of Subalgebras ◽

Irreducible Elements ◽

Algebraic Lattices ◽

Strongly Irreducible ◽

Necessary And Sufficient

α-Irreducible and α-Strongly Irreducible Ideals of a ring have been characterized in [2] and [4]. A complete lattice which is generated by compact elements is called an algebraic lattice for the simple reason that every such lattice is isomorphic to the lattice of subalgebras of a suitable universal algebra and vice-versa. In this paper, we characterize the irreducible elements and strongly irreducible elements in an algebraic lattice, which extends the results in [4] to arbitrary algebraic lattices. Also we obtain certain necessary and sufficient conditions, in terms of irreducible elements, for an algebraic lattice to satisfy the complete distributivity.

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On Malcev algebras in which all subideals are ideals

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500020886 ◽

1985 ◽

Vol 101 (3-4) ◽

pp. 353-363 ◽

Cited By ~ 3

Author(s):

Alberto Elduque

Keyword(s):

Lattice Of Subalgebras ◽

Complemented Lattice

SynopsisMalcev algebras in which the relation of being an ideal is transitive are studied as well as those Malcev algebras in which every subalgebra satisfies that condition. These algebras are closely related to those in which right multiplication by any element is semisimple and they are used to determine Malcev algebras with a relatively complemented lattice of subalgebras.

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