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<p><i>L</i>-algebras form an algebraic structure which arises, for instance, in algebraic logic, group theory, topology, and quantum theory. Examples are the projection lattice of a von Neumann algebra, the open sets of a topological space, measure algebras, or the generating <i>L</i>-algebra of an Artin-Tits group. With respect to direct products and up to isomorphism, it is proved that all indecomposable <i>L</i>-algebras are cancellable. An additive version of a cancellation result of Banaschewski and Lowen (Proc. AMS, 2004) is obtained as a special case. Extending a well-known concept for Artin-Tits groups and von Neumann algebras, the centre of a bounded <i>L</i>-algebra <i>X</i> is introduced and shown to be a Boolean subalgebra reflecting the possible decompositions of <i>X</i>. Simple descriptions of the centre are obtained in special cases.</p>

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Decomposition of L-algebras

  • 期刊论文

  • 发表日期：

    2025年12月23日

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  • Wolfgang Rump

摘要

L-algebras form an algebraic structure which arises, for instance, in algebraic logic, group theory, topology, and quantum theory. Examples are the projection lattice of a von Neumann algebra, the open sets of a topological space, measure algebras, or the generating L-algebra of an Artin-Tits group. With respect to direct products and up to isomorphism, it is proved that all indecomposable L-algebras are cancellable. An additive version of a cancellation result of Banaschewski and Lowen (Proc. AMS, 2004) is obtained as a special case. Extending a well-known concept for Artin-Tits groups and von Neumann algebras, the centre of a bounded L-algebra X is introduced and shown to be a Boolean subalgebra reflecting the possible decompositions of X. Simple descriptions of the centre are obtained in special cases.