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Non-modular and non-distributive primitive ordered subsets of lattices. (English) Zbl 0796.06009

Let \(L\) be a lattice and \(P\) a finite non-void subset of \(L\). Then \(P\) is called a primitive subset of \(L\) if \(\bigwedge(\theta(x,x\lor y)\); \(x,y\in P\), \(x\neq x\lor y)\neq \omega\), where \(\omega\) is the trivial congruence relation on \(L\) and \(\theta(x,y)\) denotes the principal congruence relation on \(L\) generated by the pair \((x,y)\).
Lattices not containing an isomorphic copy of a member from a given set of finite ordered sets as a primitive subset form a variety of lattices. The author characterizes the variety of distributive lattices by some collections of primitive ordered sets. Finally, two small non- distributive varieties of lattices are described similarly.
Reviewer: B.F.Šmarda (Brno)

MSC:

06B20 Varieties of lattices
06A99 Ordered sets
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References:

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