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Keywords:
order affine completeness; congruences of lattices; tolerances of lattices
order affine completeness; congruences of lattices; tolerances of lattices
Summary:
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 $.
References:
[1] I. Chajda, B. Zelinka: Tolerance relation on lattices. Čas. Pěst. Mat. 99 (1974), 394–399. MR 0360380
[2] K. Kaarli: On varieties generated by functionally complete algebras. Algebra Univers. 29 (1992), 495–502. DOI 10.1007/BF01190776 | MR 1201174 | Zbl 0774.08003
[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 10.1023/A:1024643815068 | MR 1984855
[4] P. Hegedűs, P. P. Pálfy: Finite modular congruence lattices. Algebra Univers. 54 (2005), 105–120. MR 2217968
[5] K. Kaarli, A. F. Pixley: Polynomial Completeness in Algebraic Systems. Chapman & Hall/CRC, Boca Raton, 2000. MR 1888967
[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 0537415 | Zbl 0403.06004
[7] D. Schweigert: Über die endliche, ordnungspolynomvollständige Verbände. Monatsh. Math. 78 (1974), 68–76. DOI 10.1007/BF01298196 | MR 0340124
[8] R. Wille: Eine Characterisierung endlicher, ordnungspolynomvollständiger Verbände. Arch. Math. (Basel) 28 (1977), 557–560. DOI 10.1007/BF01223966 | MR 0447062
[9] R. Wille: Über endliche, ordnungaffinvollständige Verbände. Math. Z. 155 (1977), 103–107. DOI 10.1007/BF01214209 | MR 0485604
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