Geodesic
Order affine completeness of lattices with Boolean congruence lattices
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 4, pp. 1049-1065 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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This paper grew out from attempts to determine which modular lattices of finite height are locally order affine complete. A surprising discovery was that one can go quite far without assuming the modularity itself. The only thing which matters is that the congruence lattice is finite Boolean. The local order affine completeness problem of such lattices ${\mathbf L}$ easily reduces to the case when ${\mathbf L}$ is a subdirect product of two simple lattices ${\mathbf L}_1$ and ${\mathbf L}_2$. Our main result claims that such a lattice is locally order affine complete iff ${\mathbf L}_1$ and ${\mathbf L}_2$ are tolerance trivial and one of the following three cases occurs: 1) ${\mathbf L}={\mathbf L}_1\times {\mathbf L}_2$, 2) ${\mathbf L}$ is a maximal sublattice of the direct product, 3) ${\mathbf L}$ is the intersection of two maximal sublattices, one containing $\langle 0,1\rangle $ and the other $\langle 1,0\rangle $.
This paper grew out from attempts to determine which modular lattices of finite height are locally order affine complete. A surprising discovery was that one can go quite far without assuming the modularity itself. The only thing which matters is that the congruence lattice is finite Boolean. The local order affine completeness problem of such lattices ${\mathbf L}$ easily reduces to the case when ${\mathbf L}$ is a subdirect product of two simple lattices ${\mathbf L}_1$ and ${\mathbf L}_2$. Our main result claims that such a lattice is locally order affine complete iff ${\mathbf L}_1$ and ${\mathbf L}_2$ are tolerance trivial and one of the following three cases occurs: 1) ${\mathbf L}={\mathbf L}_1\times {\mathbf L}_2$, 2) ${\mathbf L}$ is a maximal sublattice of the direct product, 3) ${\mathbf L}$ is the intersection of two maximal sublattices, one containing $\langle 0,1\rangle $ and the other $\langle 1,0\rangle $.
Classification : 06B10, 08A30, 08A40
Keywords: order affine completeness; congruences of lattices; tolerances of lattices 
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     author = {Kaarli, Kalle and Kuchmei, Vladimir},
     title = {Order affine completeness of lattices with {Boolean} congruence lattices},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1049--1065},
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     zbl = {1174.06304},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2007_57_4_a0/}
}
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Kaarli, Kalle; Kuchmei, Vladimir. Order affine completeness of lattices with Boolean congruence lattices. Czechoslovak Mathematical Journal, Tome 57 (2007) no. 4, pp. 1049-1065. http://geodesic.mathdoc.fr/item/CMJ_2007_57_4_a0/

[1] I.  Chajda, B.  Zelinka: Tolerance relation on lattices. Čas. Pěst. Mat. 99 (1974), 394–399. | MR

[2] K. Kaarli: On varieties generated by functionally complete algebras. Algebra Univers. 29 (1992), 495–502. | DOI | MR | Zbl

[3] G.  Czédli, E.  Horváth, and S.  Radeleczki: On tolerance lattices of algebras in congruence modular lattices. Acta Math. Hung. 100 (2003), 9–17. | DOI | MR

[4] P. Hegedűs, P.  P.  Pálfy: Finite modular congruence lattices. Algebra Univers. 54 (2005), 105–120. | MR

[5] K.  Kaarli, A.  F.  Pixley: Polynomial Completeness in Algebraic Systems. Chapman & Hall/CRC, Boca Raton, 2000. | MR

[6] M.  Kindermann: Über die Äquivalenz von Ordnungspolynomvollständigkeit und Toleranzeinfachheit endlicher Verbände. Contributions to General Algebra (Proc. Klagenfurt Conf. 1978), Verlag J.  Heyn, Klagenfurt, 1979, pp. 145–149. | MR | Zbl

[7] D.  Schweigert: Über die endliche, ordnungspolynomvollständige Verbände. Monatsh. Math. 78 (1974), 68–76. | DOI | MR

[8] R.  Wille: Eine Characterisierung endlicher, ordnungspolynomvollständiger Verbände. Arch. Math. (Basel) 28 (1977), 557–560. | DOI | MR

[9] R.  Wille: Über endliche, ordnungaffinvollständige Verbände. Math.  Z. 155 (1977), 103–107. | DOI | MR