Another independent self-dual basis for the trivial variety (Q2475559)

The present paper deals with algebras whose similarity type \(\tau\) consists of two binary operation symbols \(\vee\) and \(\wedge\), called join and meet. The dual of a term \(p\) of type \(\tau\), denoted by \(\widetilde p\), is obtained by interchanging all its meet and join symbols. A set \(\Sigma\) of identities of type \(\tau\) is called self-dual if, for each identity \(p=q\) in \(\Sigma\), the identity \(\widetilde p=\widetilde q\) is in \(\Sigma\), and a variety of type \(\tau\) is called self-dual if its set of valid identities is self-dual. In a former paper by the same authors Algebra Univers. 52, 501--517 (2004; Zbl 1084.06004)], it was shown that each finitely based self-dual variety of modular lattices has an independent self-dual 3-basis (i.e., an irredundant self-dual equational basis with 3 identities). For any such nontrivial variety, this basis was constructed in a uniform manner, whereas the trivial variety required a different construction. In the present paper, the authors give a new independent self-dual 3-basis for the trivial lattice variety. The main feature of this new basis is the fact that the identity \(x=y\) follows more easily from it that from the corresponding self-dual 3-basis in the former paper.