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Generators of existence varieties of regular rings and complemented Arguesian lattices

Christian Herrmann, Marina Semenova (2010)

Open Mathematics

We proved in an earlier work that any existence variety of regular algebras is generated by its simple unital Artinian members, while any existence variety of Arguesian sectionally complemented lattices is generated by its simple members of finite length. A characterization of the class of simple unital Artinian members [members of finite length, respectively] of such varieties is given in the present paper.

Orthomodular lattices that are horizontal sums of Boolean algebras

Ivan Chajda, Helmut Länger (2020)

Commentationes Mathematicae Universitatis Carolinae

The paper deals with orthomodular lattices which are so-called horizontal sums of Boolean algebras. It is elementary that every such orthomodular lattice is simple and its blocks are just these Boolean algebras. Hence, the commutativity relation plays a key role and enables us to classify these orthomodular lattices. Moreover, this relation is closely related to the binary commutator which is a term function. Using the class of horizontal sums of Boolean algebras, we establish an identity which...

Representation of algebraic distributive lattices with ℵ1 compact elements as ideal lattices of regular rings.

Friedrich Wehrung (2000)

Publicacions Matemàtiques

We prove the following result: Theorem. Every algebraic distributive lattice D with at most ℵ1 compact elements is isomorphic to the ideal lattice of a von Neumann regular ring R.(By earlier results of the author, the ℵ1 bound is optimal.) Therefore, D is also isomorphic to the congruence lattice of a sectionally complemented modular lattice L, namely, the principal right ideal lattice of R. Furthermore, if the largest element of D is compact, then one can assume that R is unital, respectively,...

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