Bounded lattice structured discriminator varieties (Q616628)

The paper deals with bounded lattice-structured discriminator varieties, i.e., discriminator varieties \(\mathcal V\) interpreting the variety \(\mathcal{L}_{01}\) of bounded lattices. The main results are as follows: {\parindent4mm \begin{itemize}\item[{\(\bullet\)}] Discriminator varieties \(\mathcal V\) are characterized by equational conditions on a binary term. \item[{\(\bullet\)}] A discriminator variety interpreting \(\mathcal{L}_{01}\) which is interpretable in every \(\mathcal V\) is described. \item[{\(\bullet\)}] Four special unary terms for a variety \(\mathcal V\) are derived from a special binary term: a term defining an existencial quantifier, a term defining a universal quantifier, a term defining a weak pseudocomplementation, and a term defining a dual weak pseudocomplementation. All these terms determine in any algebra the same subreduct which is a Boolean algebra. \item[{\(\bullet\)}] The Boolean subreduct \(B(\mathbb A)\) of an algebra \(\mathbb{A}\in \mathcal V\) is isomorphic to the lattice of principal congruences on \(\mathbb A\), and it determines all congruences of \(\mathbb A\). \item[{\(\bullet\)}] A term operation of weak implication for any variety \(\mathcal V\) is considered and the deductive systems relative to this operation are characterized. \end{itemize}}