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Abstract: Algebraic properties of lattices of quotients of finite posets are considered. Using the known duality between the category of all finite posets together with all order-preserving maps and the category of all finite distributive (0; 1)-lattices together with all (0; 1)-lattice homomorphisms, algebraic and arithmetic properties of maximal proper sublattices and, in particular, Frattini sublattices of finite distributive (0; 1)-lattices are thereby obtained. Addresses: Dept. of Math. and Comp.... (Update)
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...1. Three lattices witnessing the ambiguity of Theorem 1 In the case of finite distributive lattices, we prove a converse of Theorem 1: [3] Sub L for finite distributive lattices L 3 Theorem 2. Let L be a finite distributive lattice, let K # Sub L, and let x # L K with K #...
...subgroup hold more generally in any algebraic structure. Numerous papers have discussed Frattini sublattices. For a sample, see [1, 2, 8, 21]. There are also papers studying the concept in the context of Moufang loops [13] closure algebras [24] Stone algebras [7]...
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2: Maximal sublattices of finite distributive lattices (context) - Rival - 1973
2: Deciding Frattini is NP-complete (context) - Ch - 1994
2: Maximal sublattices and Frattini sublattices of bounded lattices - Adams, Freese et al.
BibTeX entry: (Update)
M. E. Adams, P. Dwinger and J. Schmid, `Maximal sublattices of finite distributive lattices', Algebra Universalis 36 (1996) 488--504. http://citeseer.ist.psu.edu/adams96maximal.html More
@misc{ adams96maximal,
author = "M. Adams and P. Dwinger and J. Schmid",
title = "Maximal sublattices of finite distributive lattices",
text = "M. E. Adams, P. Dwinger and J. Schmid, `Maximal sublattices of finite distributive
lattices', Algebra Universalis 36 (1996) 488--504.",
year = "1996",
url = "citeseer.ist.psu.edu/adams96maximal.html" }
Citations (may not include all citations):23 University of Missouri Press (context) - Balbes - 1974
14 Dimension theory for ordered sets (context) - Kelly, Trotter - 1982
12 Maximal sublattices of finite distributive lattices (context) - Rival - 1973
6 Maximal sublattices and Frattini sublattices of bounded latt.. - Adams, Freese et al.
5 The Frattini sublattice of a finite distributive lattice (context) - Abad, Adams - 1994
5 The Frattini sublattice of a distributive lattice (context) - Adams - 1973
3 The rank of a distributive lattice (context) - Rabinovitch, Rival - 1979
3 the Frattini sublattice of a lattice (context) - Koh - 1971
3 Deciding Frattini is NP-complete (context) - Ch - 1994
3 Posets with large dimension and relatively few critical pair.. (context) - Fishburn, Trotter - 1993
3 the Frattini sublattice of a finite distributive lattice (context) - Chen, Koh et al. - 1975
2 the complexity of interval orders and semiorders (context) - Faigle - 1987
2 Ideal theory for lattices (context) - Hashimoto - 1952
2 Frattini sublattices of distributive lattices (context) - Chen, Koh et al. - 1973
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