Standard Bases of a Vector Space Over a Linearly Ordered Incline

Abstract

Inclines are additively idempotent semirings, in which the partial order ≤ : x ≤ y if and only if x + y = y is defined and products are less than or equal to either factor. Boolean algebra, max-min fuzzy algebra, and distributive lattices are examples of inclines. In this article, standard bases of a finitely generated vector space over a linearly ordered commutative incline are studied. We obtain that if a standard basis exists, then it is unique. In particular, if the incline is solvable or multiplicatively-declined or multiplicatively-idempotent (i.e., a chain semiring), further results are obtained, respectively. For a chain semiring a checkable condition for distinguishing if a basis is standard is given. Based on the condition an algorithm for computing the standard basis is described.

Key Words:

• Basis
• Incline
• Semiring
• Standard basis
• Vector space

2000 Mathematics Subject Classification:

• 06F99

ACKNOWLEDGMENTS

This work is supported by the Innovation Program of Shanghai Municipal Education Commission (09YZ239) and the Scientific Research Foundation of Tianjin Education Committee (20070402).

Notes

Communicated by J. Zhang.