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Congruence variety; relative congruence; ordered algebra; von Neumann frame; lattice identity
For a class $K$ of structures and $A\in K$ let ${Con}^*(A)$ resp. ${Con}^{K}(A)$ denote the lattices of $*$-congruences resp. $K$-congruences of $A$, cf. Weaver [25]. Let ${Con}^*(K):=I\lbrace {Con}^*(A)\colon\ A \in K\rbrace $ where $I$ is the operator of forming isomorphic copies, and ${Con}^r(K):=I\lbrace {Con}^{K}(A)\colon\ A \in K\rbrace $. For an ordered algebra $A$ the lattice of order congruences of $A$ is denoted by ${Con}^{<}(A)$, and let ${Con}^{<}(K):=I\lbrace {Con}^{<}(A)\colon\ A \in K\rbrace $ if $K$ is a class of ordered algebras. The operators of forming subdirect squares and direct products are denoted by $Q^s$ and $P$, respectively. Let $\lambda $ be a lattice identity and let $\Sigma $ be a set of lattice identities. Let $\Sigma \mathrel {\models _c}\lambda\ (r;Q^s,P)$ denote that for every class $K$ of structures which is closed under $Q^s$ and $P$ if $\Sigma $ holds is ${Con}^r(K)$ then so does $\lambda$. The consequence relations $\Sigma \mathrel {\models _c}\lambda\ (*;Q^s)$,   $\Sigma \mathrel {\models _c}\lambda\ (\le ;Q^s)$ and $\Sigma \mathrel {\models _c}\lambda\ (H,S,P)$ are defined analogously; the latter is the usual consequence relation in congruence varieties (cf. Jónsson [19]), so it will also be denoted simply by $\mathrel {\models _c}$. If $\Sigma \lnot \models \lambda $ (in the class of all lattices) then the above-mentioned consequences are called nontrivial. The present paper shows that if $\Sigma \models$ modularity and $\Sigma \mathrel {\models _c}\lambda $ is a known result in the theory of congruence varieties then $\Sigma \mathrel {\models _c}\lambda\ (*; Q^s)$, $\Sigma \mathrel {\models _c}\lambda\ (\le ;Q^s)$ and $\Sigma \mathrel {\models _c}\lambda\ (r;Q^s,P)$ as well. In most of these cases $\lambda $ is a diamond identity in the sense of [3].
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