Congruence join semidistributivity is equivalent to a congruence identity.
A lattice is join semidistributive if it satisfies the implication \[ \alpha \vee \beta = \alpha \vee \gamma \Rightarrow \alpha \vee \beta = \alpha \vee (\beta \wedge \gamma ). \] For \(\alpha ,\beta ,\gamma \in \text{Con}{\mathcal A}\) define recursively \(\beta ^0 =\beta , \, \gamma ^0 =\gamma , \,\beta ^{n+1} =\beta \wedge (\alpha \vee \gamma ^n), \, \gamma ^{n+1} = \gamma \wedge ( \alpha \vee \beta ^n)\). The main result: Let \({\mathcal V}\) be a locally finite variety. Then \({\mathcal V}\) is congruence join semidistributive if for some integer \(n\), \({\mathcal V}\) satisfies \[ \alpha \vee (\beta \wedge \gamma ) = (\alpha \vee \beta ^n) \wedge (\alpha \vee \gamma ^n) \] as a congruence identity. This identity is strong enough to force join semidistributivity in any lattice.
scientific article; zbMATH DE number 2158750
scientific article