PosetElements

The elements of a finite poset, in a linear extension — the order every other head here indexes values by.

PosetElements(poset)the elements, listed so that smaller ones come first

Typed signatures

Type	Package	Applies
`(expression<BooleanLattice> \| expression<Chain> \| expression<DivisorLattice>) -> list<list<integer> \| number>`	`incidence`	always

Domain: Incidence algebras

Details

The linear extension is what makes $\zeta$ upper-triangular with ones on the diagonal, hence invertible over the integers — and therefore what makes $\mu$ integral
DivisorLattice elements are integers; BooleanLattice elements are subsets; Chain elements are $1 \dots n$
AlgebraDimension(IncidenceAlgebra(poset)) counts the intervals: $\binom{n+1}{2}$ for a chain, $3^n$ for the Boolean lattice

Examples

Basic #

\mathrm{PosetElements}(\mathrm{DivisorLattice}(12))

Out

\bigl\lbrack1, 2, 3, 4, 6, 12\bigr\rbrack

\mathrm{PosetElements}(\mathrm{Chain}(4))

Out

\bigl\lbrack1, 2, 3, 4\bigr\rbrack

Properties #

$3^3$ intervals

\mathrm{AlgebraDimension}(\mathrm{IncidenceAlgebra}(\mathrm{BooleanLattice}(3)))

Out

27

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