Séminaire Lotharingien de Combinatoire, 91B.6 (2024), 12 pp.
Sam Hopkins
Upho Lattices and Their Cores
Abstract.
A poset is called upper homogeneous, or "upho," if every principal
order filter is isomorphic to the original poset. We study enumerative
and structural properties of (finite type N-graded) upho
posets. The first important observation we make about upho posets is
that their rank generating functions and characteristic generating
functions are multiplicative inverses of one another. This means that
each upho lattice has associated to it a finite graded lattice, called
its core, which determines its rank generating function. We
investigate which finite graded lattices arise as cores of upho
lattices, providing both positive and negative results. On the one
hand, we show that many well-studied finite lattices do arise as
cores, and we present combinatorial and algebraic constructions of the
upho lattices into which they embed. On the other hand, we show there
are obstructions which prevent many finite lattices from being cores.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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