Dual discriminator subvarieties of a variety
The dual discriminator function \(d(x,y,z)\) on a set \(A\) is defined as \(d (a, b, c) = a\) if \(a = b\) and \(d(a,b,c) = c\) if \(a \neq b\). Let \(q(x,y,z)\) be a term of a variety \({\mathcal V}\) of algebras. Then the subvariety \({\mathcal X}\) of \({\mathcal V}\) generated by all algebras in \({\mathcal V}\) where \(q(x,y,z)\) yields the dual discriminator function is called the dual discriminator subvariety of \({\mathcal V}\) defined by \(q(x,y,z)\). A dual discriminator subvariety of \({\mathcal V}\) defined by some term is called a dual discriminator subvariety of \({\mathcal V}\). For example, the variety \({\mathcal L}\) of all lattices contains two dual discriminator subvarieties: the variety \({\mathcal D}\) of distributive lattices and the variety \({\mathcal O}\) consisting of the trivial lattice, only. An algebra \(\langle A; \vee, \wedge \rangle\) is called a weakly associative lattice (WAL), if the two binary operations satisfy the following identities: \[ \begin{aligned} & x \vee x = x = x \wedge x, \\ & x \vee y = y \vee x \text{ and } x \wedge y = y \wedge x, \\ & x \vee (y \wedge x) = x = x \wedge (y \vee x), \\ & x \wedge \bigl[ (x \vee y) \wedge (x \vee z) \bigr] = x = x \vee \bigl[ (x \wedge y) \vee (x \wedge z) \bigr]. \end{aligned} \] Let \({\mathcal W}\) be the variety of all WALs. Since the number of ternary WAL-terms is countable, the number of dual discriminator subvarieties of \({\mathcal W}\) can only be countable. The question is whether this number is finite or infinite. In the paper, infinitely many terms in \({\mathcal W}\) such that they define distinct dual discriminator subvarieties of \({\mathcal W}\) are constructed. As a supplement, a variety \({\mathcal V}\) of WALs which is the union of its dual discriminator subvarieties \({\mathcal V}_0 \subset {\mathcal V}_1 \subset \cdots\) and is not a dual discriminator variety is presented.
number of dual discriminator subvarieties
scientific article; zbMATH DE number 836827