Reducible classes of finite lattices. (English) Zbl 0891.06001
Authors’ summary: We study a notion of reducibility in finite lattices. An element \(x\) of a (finite) lattice \(L\) satisfying certain properties is deletable if \(L-x\) is a lattice satisfying the same properties. A class of lattices is reducible if each lattice of this class admits (at least) one deletable element (equivalently if one can go from any lattice in this class to the trivial lattice by a sequence of lattices of the class obtained by deleting one element in each step). First we characterize the deletable elements in a pseudocomplemented lattice what allows to prove that the class of pseudocomplemented lattices is reducible. Then we characterize the deletable elements in semimodular, modular and distributive lattices what allows to prove that the classes of semimodular and locally distributive lattices are reducible. In conclusion the notion of reducibility for a class of lattices is compared with some other notions like the notion of order variety.
Reviewer: G.A.Grätzer (Winnipeg)
MSC:
| 06B05 | Structure theory of lattices |
| 06C10 | Semimodular lattices, geometric lattices |
| 06D15 | Pseudocomplemented lattices |
| 06C05 | Modular lattices, Desarguesian lattices |
| 06D05 | Structure and representation theory of distributive lattices |
Keywords:
dismantlable; semimodular lattices; modular lattices; reducibility; finite lattices; deletable element; pseudocomplemented lattices; locally distributive lattices; order varietyReferences:
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