Solution to Problems of Schmidt and Quackenbush from 1974 and 1985: Tensor Products of Semilattices
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چکیده
If M is a finite complemented modular lattice with n atoms and D is a bounded distributive lattice, then the Priestley power M [D] is shown to be isomorphic to the poset of normal elements of Dn, thus solving a problem of E. T. Schmidt from 1974. It is shown that there exist a finite modular lattice A not having M4 as a sublattice and a finite modular lattice B such that A⊗B is not semimodular, thus refuting a conjecture of Quackenbush from 1985. It is shown that the tensor product of M3 with a finite modular lattice B is supersolvable if and only if B is distributive, thus proving a conjecture of Quackenbush from 1985.
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