FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2011/2012, VOLUME 17, NUMBER 5, PAGES 129-146

The atomic theory of left division of two-sided ideals of semirings with unit

A. E. Pentus
M. R. Pentus

Abstract

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We consider two-sided ideals of semirings with unit. We study the theory of two-sided ideals in the signature consisting of the binary predicate symbol $$Í and a binary function symbol that denotes the left division of ideals. We prove the polynomial-time decidability of the problem of deciding whether a given atomic formula in this signature is valid for all semirings with unit and all valuations. A similar result holds for atomic formulas with the right division, but without the left division.

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Last modified: October 18, 2012