Varieties generated by modes of submodes
Any operation on a set \(A\) may be lifted up to an operation on the set of non-empty subsets of \(A\) in a natural way. In this way, for any algebra \((A,\Omega)\) one obtains its power algebra of subsets of \(A\). G. Grätzer and H. Lakser proved that for a variety \(V\) of algebras, the variety \(V\Sigma\) generated by the power algebras of algebras in \(V\) satisfies precisely the consequences of the linear identities true in \(V\). If \(V\) is a variety of idempotent and entropic algebras, then the set of non-empty subalgebras of any algebra in \(V\) forms a subalgebra of its power algebra. In this way, for any algebra \((A,\Omega)\) in \(V\), one obtains its algebra of subalgebras. The authors solve a long-standing problem concerning identities satisfied in the variety \(VS\) generated by the algebras of subalgebras of algebras in any idempotent and entropic variety. They prove that if \(V\) is idempotent and entropic and \(V\Sigma\) is locally finite, then the variety \(VS\) is defined by the idempotent and linear identities true in \(V\).
scientific article; zbMATH DE number 6132824
scientific article