Geodesic
ON DIRECT DECOMPOSITIONS OF CERTAIN ORTHOMODULAR LATTICES
Acta mathematica Universitatis Comenianae, Tome 60 (1991) no. 1 
P. Kon\opka; S. Pulmannova. ON DIRECT DECOMPOSITIONS OF CERTAIN ORTHOMODULAR LATTICES. Acta mathematica Universitatis Comenianae, Tome 60 (1991) no. 1. http://geodesic.mathdoc.fr/item/AMUC_1991_60_1_a10/
@article{AMUC_1991_60_1_a10,
     author = {P. Kon\opka and S. Pulmannova},
     title = {ON {DIRECT} {DECOMPOSITIONS} {OF} {CERTAIN} {ORTHOMODULAR} {LATTICES}},
     journal = {Acta mathematica Universitatis Comenianae},
     year = {1991},
     volume = {60},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/AMUC_1991_60_1_a10/}
}
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Voir la notice de l'article provenant de la source Comenius University

Let $L$ be an orthomodular lattice. For $a,b\in L$ define $a\pl^cb$ if either $a$ and $b$ both belong to the centre $C(L)$ of $L$ or if $\a,b\\cap C(L)=\emptyset$ and $a\pl b$ (i.e. $a$ is compatible with $b$). Let $R$ be the transitive closure of the relation $\pl^c$. Then there exist at least three equivalence classes of the relation $R$ in $L$ if and only if either $L$ is a horizontal sum (if $C(L)=\0,1\$) or $L$ is a direct product of a Boolean algebra and a horizontal sum.