Abstract
In a natural way, we can “lift” any operation defined on a set A to an operation on the set of all non-empty subsets of A and obtain from any algebra (\({A, \Omega}\)) its power algebra of subsets. G. Grätzer and H. Lakser proved that for a variety \({\mathcal{V}}\), the variety \({\mathcal{V}\Sigma}\) generated by power algebras of algebras in \({\mathcal{V}}\) satisfies precisely the consequences of the linear identities true in \({\mathcal{V}}\). For certain types of algebras, the sets of their subalgebras form subalgebras of their power algebras. They are called the algebras of subalgebras. In this paper, we partially solve a long-standing problem concerning identities satisfied by the variety \({\mathcal{VS}}\) generated by algebras of subalgebras of algebras in a given variety \({\mathcal{V}}\). We prove that if a variety \({\mathcal{V}}\) is idempotent and entropic and the variety \({\mathcal{V}\Sigma}\) is locally finite, then the variety \({\mathcal{VS}}\) is defined by the idempotent and linear identities true in \({\mathcal{V}}\).
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Presented by K. Kearnes.
Dedicated to Professor Anna Romanowska
While working on this paper, the authors were supported by the Statutory Grant of Warsaw University of Technology 504G/1120/0087/000.
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Pilitowska, A., Zamojska-Dzienio, A. Varieties generated by modes of submodes. Algebra Univers. 68, 221–236 (2012). https://doi.org/10.1007/s00012-012-0201-4
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DOI: https://doi.org/10.1007/s00012-012-0201-4