In this paper, we give a consistent presentation of dimension properties and properties of bases for vector spaces over distributive lattices. The bases consisting of join irreducible vectors are studied and their uniqueness is proved. Criteria for the following are proved: the set of join irreducible vectors is a generating set for a vector space; this set is a vector space basis; all bases contain the same number of vectors. A criterion of uniqueness for the basis is proved. The basis containing the greatest number of vectors is found. We give a description for all standard bases of a vector space. We prove a theorem allowing one to calculate the space dimension and to find the basis of the smallest number of vectors by known algorithms. These results are applied to vector spaces over chains: we prove that there exists a standard basis, that the basis of join irreducible vectors is the standard basis, that a standard basis is unique. We calculate the dimension of the arithmetic space and describe all bases containing the smallest number of vectors. It is proved that all such bases are standard.
Characteristics for lattice matrices to be prime and existence criteria for prime matrices over distributive lattices are obtained.
PreliminariesA partially ordered set (Q, ≤) is called a ≤-antichain if a ≤ b for any a, b ∈ Q. Let (P, ∧, ∨, ≤) be a lattice. Any matrix with entries in P is called a lattice matrix (or a matrix over the lattice). A matrix over a Boolean lattice is a Boolean matrix, a matrix over the two-element Boolean lattice is a { 0, 1}-Boolean matrix, and a matrix over the fuzzy lattice is a fuzzy matrix.Denote by P m×n the set of all (m × n)-lattice matrices, where n, m ≥ 1. Let (P, ∧, ∨, ≤) be a lattice with a zero 0 and a one 1. Then define the identity matrix E = E n×n ∈ P n×n byAn (m×n)-matrix with all entries 0 is called a null matrix and is denoted by 0 = 0 m×n . An (m×n)-matrix with all entries 1 is called a universal matrix and is denoted by J = J m×n .Let A ∈ P m×n , B ∈ P n×k , and C ∈ P m×k . The product C = A • B is defined as follows:
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