## Congruence normality: The characterization of the doubling class of convex sets.(English)Zbl 0804.06006

Let $$D= (D_ L)$$, where $$L$$ ranges over the class of finite lattices and $$D_ L$$ is a system of convex subsets of $$L$$. A finite lattice, $$M$$, is said to be constructible from $$D$$ by doubling if there is a finite sequence of finite lattices $$1= L_ 0, L_ 1, \dots, L_ n =M$$ and convex sets $$C_ i \subseteq L_ i$$ with $$C_ i\in D_{L_ i}$$, $$i<n$$, such that $$L_{i+1} \cong L_ i [C_ i]$$. The doubling class $$D$$ is the class of all finite lattices that are constructible from $$D$$ by doubling: it is denoted by $$1[D]$$.
The author characterizes the doubling classes of several classes as: the class $$CC$$ of all connected convex sets of finite lattices, the classes $$LC$$ of all lower and $$UC$$ all upper pseudo-intervals, and the class $$INT$$ of all the intervals, where a lower pseudo-interval in $$L$$ in a (finite) lattice is a member of $$C_ L$$ with $$\bigwedge_ L V\in V$$ and an upper pseudo-interval is defined dually. An interval in $$L$$ is a member of $$C_ L$$ of the form $$u\setminus v$$. The characterizations obtained are, respectively: the class $$CN$$ of all congruence normal finite lattices, the classes $$CU_ \vee$$, $$CU_ \wedge$$ and $$CU$$ of all join resp. meet resp. fully congruence uniform lattices. A comparison with a paper of W. Geyer written in the language of concept lattices is made.

### MSC:

 06B10 Lattice ideals, congruence relations 06B20 Varieties of lattices
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### References:

 [1] Day, A.,A simple solution of the word problem for lattices, Can. Math. Bull.13 (1970), 253-254. · Zbl 0206.29702 [2] Day, A.,Characterizations of finite lattices that are bounded-homomorphic images or sublattices of free lattices. Can. J. Math.31 (1979), 69-78. · Zbl 0432.06007 [3] Day, A.,Doubling constructions in lattice theory, Can. J. Math.44 (1992), 252-269. · Zbl 0768.06006 [4] Freese, R. andNation, J. B.,Covers in free lattices, Trans. Amer. Math. Soc.288 (1985), 1-42. · Zbl 0567.06008 [5] Geyer, W.,The generalized doubling construction and formal concept analysis, Algebra Universalis, to appear. · Zbl 0829.06007
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