Abstract
The relationship of projectivity between two quotients in a lattice is shown not to be first-order definable, in any nondistributive lattice variety. The proof depends on a special kind of subdirect power construction that shows the existence of arbitrarily long non-shortenable projectivities in such a variety. A similar result holds for weak projectivities. Even so, weak projectivities of bounded length do suffice to determine principal congruences in any variety generated by a finite lattice.
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Dedicated to Garrett Birkhoff
Author's research was supported in part by NSF Grant MCS 81-02519.
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Baker, K.A. Nondefinability of projectivity in lattice varieties. Algebra Universalis 17, 267–274 (1983). https://doi.org/10.1007/BF01194536
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DOI: https://doi.org/10.1007/BF01194536