Algebra universalis

, Volume 68, Issue 3–4, pp 221–236 | Cite as

Varieties generated by modes of submodes

Open Access
Article

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

2010 Mathematics Subject Classification

Primary: 03C05 Secondary: 03G25 08A30 08A62 06E25 06A12 08B99 

Key words and phrases

idempotent entropic modes power algebras congruence relations identities varieties of finitary algebras locally finite varieties 

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© The Author(s) 2012

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

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

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