Proctor and Scoppetta conjectured that
(1) there exists an infinite locally finite poset that satisfies their conditions VT and NTC but not SIS;
(2) there exists an infinite locally finite poset satisfying their conditions D3-C and D3MF but not both VT and FT; and
(3) there exists an infinite locally finite poset satisfying their conditions D3-C and D3MD but not NCC.
In this note, the conjecture of Proctor and Scoppetta, which is related to d-complete posets, is proven.
Birkhoff, G. Lattice Theory (third edition). American Mathematical Society, Providence, Rhode Island, 1967.
Davey, B. A., and Priestley, H. A. Introduction to Lattices and Order (second edition). Cambridge University Press, Cambridge, 2002.
Farley, J. D. Cancel Culture: The Search for Universally Cancellable Exponents ofPosets. Manuscript (2021).
Proctor, R. A., and Scoppetta, L. M. d-complete Posets: Local Structural Axioms, Properties, and Equivalent Definitions. Order 36 (2019), 399-422.
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