, Volume 68, Issue 3-4, pp 221-236,
Open Access This content is freely available online to anyone, anywhere at any time.
Date: 17 Oct 2012

Varieties generated by modes of submodes

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

Presented by K. Kearnes.
While working on this paper, the authors were supported by the Statutory Grant of Warsaw University of Technology 504G/1120/0087/000.
Dedicated to Professor Anna Romanowska