Algebra universalis

, Volume 68, Issue 3, pp 221–236

Varieties generated by modes of submodes

Open AccessArticle

DOI: 10.1007/s00012-012-0201-4

Cite this article as:
Pilitowska, A. & Zamojska-Dzienio, A. Algebra Univers. (2012) 68: 221. doi:10.1007/s00012-012-0201-4


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 \({\mathcal{V}}\).

2010 Mathematics Subject Classification

Primary: 03C05Secondary: 03G2508A3008A6206E2506A1208B99

Key words and phrases

idempotententropicmodespower algebrascongruence relationsidentitiesvarieties of finitary algebraslocally finite varieties
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© The Author(s) 2012

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Authors and Affiliations

  1. 1.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarsawPoland