Chains, antichains, and complements in infinite partition lattices

TY - JOUR

T1 - Chains, antichains, and complements in infinite partition lattices

AU - Avery, James Emil

AU - Moyen, Jean-Yves

AU - Ruzicka, Pavel

AU - Simonsen, Jakob Grue

PY - 2018/6/1

Y1 - 2018/6/1

N2 - We consider the partition lattice Π (λ) on any set of transfinite cardinality λ and properties of Π (λ) whose analogues do not hold for finite cardinalities. Assuming AC, we prove: (I) the cardinality of any maximal well-ordered chain is always exactly λ; (II) there are maximal chains in Π (λ) of cardinality > λ; (III) a regular cardinal λ is strongly inaccessible if and only if every maximal chain in Π (λ) has size at least λ; if λ is a singular cardinal and μ<κ< λ≤ μκ for some cardinals κ and (possibly finite) μ, then there is a maximal chain of size < λ in Π (λ) ; (IV) every non-trivial maximal antichain in Π (λ) has cardinality between λ and 2 λ, and these bounds are realised. Moreover, there are maximal antichains of cardinality max (λ, 2 κ) for any κ≤ λ; (V) all cardinals of the form λκ with 0 ≤ κ≤ λ occur as the cardinalities of sets of complements to some partition P∈ Π (λ) , and only these cardinalities appear. Moreover, we give a direct formula for the number of complements to a given partition. Under the GCH, the cardinalities of maximal chains, maximal antichains, and numbers of complements are fully determined, and we provide a complete characterisation.

AB - We consider the partition lattice Π (λ) on any set of transfinite cardinality λ and properties of Π (λ) whose analogues do not hold for finite cardinalities. Assuming AC, we prove: (I) the cardinality of any maximal well-ordered chain is always exactly λ; (II) there are maximal chains in Π (λ) of cardinality > λ; (III) a regular cardinal λ is strongly inaccessible if and only if every maximal chain in Π (λ) has size at least λ; if λ is a singular cardinal and μ<κ< λ≤ μκ for some cardinals κ and (possibly finite) μ, then there is a maximal chain of size < λ in Π (λ) ; (IV) every non-trivial maximal antichain in Π (λ) has cardinality between λ and 2 λ, and these bounds are realised. Moreover, there are maximal antichains of cardinality max (λ, 2 κ) for any κ≤ λ; (V) all cardinals of the form λκ with 0 ≤ κ≤ λ occur as the cardinalities of sets of complements to some partition P∈ Π (λ) , and only these cardinalities appear. Moreover, we give a direct formula for the number of complements to a given partition. Under the GCH, the cardinalities of maximal chains, maximal antichains, and numbers of complements are fully determined, and we provide a complete characterisation.

KW - math.RA

KW - 06B05 (Primary), 06C15 (Secondary)

U2 - 10.1007/s00012-018-0514-z

DO - 10.1007/s00012-018-0514-z

M3 - Journal article

SN - 0002-5240

VL - 79

JO - Algebra Universalis

JF - Algebra Universalis

IS - 37

M1 - 37

ER -