Article

Algebra universalis

, Volume 68, Issue 3, pp 221-236

Open Access This content is freely available online to anyone, anywhere at any time.

Varieties generated by modes of submodes

  • Agata PilitowskaAffiliated withFaculty of Mathematics and Information Science, Warsaw University of Technology
  • , Anna Zamojska-DzienioAffiliated withFaculty of Mathematics and Information Science, Warsaw University of Technology Email author 

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