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G. Czédli’s tolerance factor lattice construction and weak ordered relations. (English) Zbl 07317154

Summary: G. Czédli proved that the blocks of any compatible tolerance \(T\) of a lattice \(L\) can be ordered in such a way that they form a lattice \(L/T\) called the factor lattice of \(L\) modulo \(T\). Here we show that the Dedekind-MacNeille completion of the lattice \(L/T\) is isomorphic to the concept lattice of the context \((L, L, R)\), where \(R\) stands for the reflexive weak ordered relation \(\leq \circ T\). Weak ordered relations constitute the generalization of the ordered relations introduced by S. Valentini. Reflexive weak ordered relations can be characterized as compatible reflexive relations \(R\subseteq L^2\) satisfying \(R= \leq\, \circ\, R\,\circ\, \leq\).

MSC:

06B15 Representation theory of lattices
06B23 Complete lattices, completions
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06B05 Structure theory of lattices
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References:

[1] Czédli, G., Factor lattices by tolerances, Acta Sci. Math. (Szeged), 44, 35-42 (1982) · Zbl 0484.06010
[2] Chajda, I.; Zelinka, B., Tolerance relations on lattices, Čas. Pĕst. Mat., 99, 394-399 (1974) · Zbl 0297.06001
[3] Grätzer, G., General Lattice Theory (1978), Berlin: Akademie, Berlin · Zbl 0385.06015
[4] Ganter, B.; Wille, R., Formal Concept Analysis: Mathematical Foundations (1999), Berlin: Springer, Berlin · Zbl 0909.06001
[5] Kaarli, K.; Radeleczki, S., Representation of integral quantales by tolerances, Algebra Universalis, 79, 1, 1-20 (2018) · Zbl 1471.06009
[6] Kaarli, K., Radeleczki, S.: Ordered relations, concept lattices and self-bonds. Journal of Multiple-Valued Logic and Soft Computing, to appear in Volume dedicated to the memory of Ivo Rosenberg. · Zbl 1471.06009
[7] Pöschel, R.; Radeleczki, S., Related structures with involution, Acta Math. Acad. Sci. Hung., 123, 1-2, 169-185 (2009) · Zbl 1199.08002
[8] Valentini, S., Representation theorems for quantales, Math. Log. Q., 40, 182-190 (1994) · Zbl 0816.06018
[9] Wille, R.; Rival, I., Restructuring lattice theory: an approach based on hierarchies of concepts, Ordered Sets, 445-470 (1982), Dordrecht: Reidel, Dordrecht
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