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The Kernels of Skeletal Congruences on a Distributive Lattice
Author(s) -
Cornish William H.
Publication year - 1978
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19780840113
Subject(s) - mathematics , congruence relation , distributive lattice , distributive property , congruence lattice problem , lattice (music) , combinatorics , distributivity , embedding , isomorphism (crystallography) , discrete mathematics , pure mathematics , computer science , crystal structure , crystallography , physics , chemistry , artificial intelligence , acoustics
Let L be a distributive lattice with 0 and C ( L ) be its lattice of congruences. The skeleton, SC ( L ), of C ( L ) consists of all those congruences which are the pseudocomplements of members of C ( L ), and is a complete BOOLEan lattice. An ideal is the kernel of a skeletal congruence if and only if it is an intersection of relative annihilator ideals, i.e. ideals of the form < r, s >j={ x ∈ L : x Δ r ≤ s } for suitable r, s ∈ L . The set KSC ( L ) of all such kernels forms an upper continuous distributive lattice and the map a ↦ ( a ={ x ∈ L : x ≤ a } is a lower regular joindense embedding of L into KSC ( L ). The relationship between SC ( L ) and KSC ( L ) leads to numerous characterizations of disjunctive and generalized BOOLEan lattices. In particular, a distributive lattice L is disjunctive (generalized Boolean) if and only if the map Θ ↦ ker Θ is a lattice‐isomorphism of SC ( L ) onto KSC ( L ), whose inverse is the map J ↦ Θ ( J )** (the map J ↦ Θ( J )). In addition, a study of KSC ( L ) leads to new simple proofs of results on the completions of special classes of lattices.