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Extension of Vector Lattice Homomorphisms
Author(s) -
Bernau S. J.
Publication year - 1986
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-33.3.516
Subject(s) - congruence lattice problem , homomorphism , distributive lattice , lattice (music) , mathematics , map of lattices , algebra homomorphism , combinatorics , integer lattice , discrete mathematics , vector space , reciprocal lattice , pure mathematics , distributive property , physics , condensed matter physics , quantum mechanics , acoustics , diffraction , half integer
Suppose that B is a complete Boolean algebra, D a distributive lattice and φ a lattice homomorphism from D 0 , a sublattice of D , into B ; then φ can be extended to a lattice homomorphism of D into B . This generalizes Sikorski's extension theorem for Boolean algebras. It also leads to a new proof that if N is a complete vector lattice, L a vector lattice, M a subvector lattice of L , and f a vector lattice homomorphism of M into N , then f can be extended to a vector lattice homomorphism, g , say, of L into N . The proof of this is constructive after applying the lattice homomorphism extension theorem, and leads to the previously unknown result that g is uniquely determined by the collection of polar subspaces, g ( x ) ⊥⊥ ( x ∈ L ). The paper concludes with a new approach to the known result that the vector lattice homomorphic extensions of f are precisely the extreme points of the set of positive extensions of f .