Inner products on semimodules

In this paper, the inner products on the semimodules over a commutative semiring are investigated. Some characterizations for orthogonal sets and standard orthogonal basis in the semimodules are given. In particular, an equivalent description is obtained for a semifield S satisfying the property tha...

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Bibliographic Details
Published in:Linear & multilinear algebra 2016-08, Vol.64 (8), p.1595-1616
Main Author: Tan, Yi-Jia
Format: Article
Language:English
Subjects:
adjoint
Adjoints
Equivalence
homomorphism
Homomorphisms
inner product
Mathematical analysis
Rings (mathematics)
semimodule
semiring
standard orthogonal basis
Vectors (mathematics)
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Summary:In this paper, the inner products on the semimodules over a commutative semiring are investigated. Some characterizations for orthogonal sets and standard orthogonal basis in the semimodules are given. In particular, an equivalent description is obtained for a semifield S satisfying the property that every standard orthogonal set in a finitely generated semimodule over S can be extended to a standard orthogonal basis for . Also, the adjoint homomorphisms of the semimodules are discussed and some properties of the adjoint homomorphisms are obtained. Partial results obtained in this paper generalize and develop corresponding results for semilinear spaces of n-dimensional vectors over commutative zerosumfree semirings and for unitary spaces.
ISSN:0308-1087
1563-5139
DOI:10.1080/03081087.2015.1112341