Relation identities in implication algebras
Paolo Lipparini · arXiv (Cornell University) · 2018
Let $\alpha$, $\beta$, $\gamma, \dots$ $\Theta$, $\Psi, \dots$ $R$, $S$, $T, \dots$ be variables for, respectively, congruences, tolerances and reflexive admissible relations. Let juxtaposition denote intersection. We show that the identity $\alpha( \beta \circ \Theta ) \subseteq \alpha \beta \circ \Theta \circ \alpha \beta$ generally fails in (the set of reflexive and admissible relations on) implication algebras. This is somewhat surprising, since implication algebras not only satisfy $\alpha( \beta \circ \gamma ) \subseteq \alpha \beta \circ \alpha \gamma \circ \alpha \beta $, which is an identity equivalent to $3$-distributivity, but do satisfy strong related identities such as $R( S \circ T \circ S ) \subseteq R S \circ RT \circ RT \circ RS$.