On a problem of G. Grätzer (Q1964735)

Recall that \(\text{Id} (K)\), for a class \(K\) of lattices, denotes the set of all identities \(e: p=q\) holding in all lattices of \(K\). An equational class \(K\) of lattices is said to be self-dual, if \(e\in \text{Id} (K)\) implies \(e^d\in \text{Id} (K)\), where \(e^d: p^d= q^d\) denotes the dual identity of \(e\). The author establishes the following result: Let \(K\) be an equational class of lattices. Then \(K\) has the property ``\(L\in K\) and \(\text{Sub} (L)\cong \text{Sub} (L_1)\) implies \(L_1\in K\)'' if and only if \(K\) is self-dual. This is a solution to problem I.5 of \textit{G. Grätzer}'s monograph [General lattice theory, Birkhäuser Verlag, Basel und Stuttgart (1978; Zbl 0385.06015)].