Diamond identities for relative congruences

Czédli, Gábor. "Diamond identities for relative congruences." Archivum Mathematicum 031.1 (1995): 65-74. <http://eudml.org/doc/247700>.

@article{Czédli1995,
abstract = {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].},
author = {Czédli, Gábor},
journal = {Archivum Mathematicum},
keywords = {Congruence variety; relative congruence; ordered algebra; von Neumann frame; lattice identity; ordered algebra; von Neumann frame; lattice identities; congruence variety; relative congruences; diamond identity; modularity},
language = {eng},
number = {1},
pages = {65-74},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Diamond identities for relative congruences},
url = {http://eudml.org/doc/247700},
volume = {031},
year = {1995},
}

TY - JOUR
AU - Czédli, Gábor
TI - Diamond identities for relative congruences
JO - Archivum Mathematicum
PY - 1995
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 031
IS - 1
SP - 65
EP - 74
AB - 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].
LA - eng
KW - Congruence variety; relative congruence; ordered algebra; von Neumann frame; lattice identity; ordered algebra; von Neumann frame; lattice identities; congruence variety; relative congruences; diamond identity; modularity
UR - http://eudml.org/doc/247700
ER -