## 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:

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