# isLexIdeal -- determine whether an ideal is a lexicographic ideal

## Synopsis

• Usage:
B=isLexIdeal I
• Inputs:
• I, an ideal, a homogeneous ideal in a polynomial ring or quotient of a polynomial ring by a homogeneous ideal
• Outputs:
• B, , true if I is a lexicographic ideal in ring I and false otherwise

## Description

Given an ideal I in a ring R that is either a polynomial ring or a quotient of a polynomial ring by a monomial ideal, isLexIdeal computes bases of I in each degree up through the maximum degree of a minimal generator of I to determine whether I is a lexicographic ideal in R.

 i1 : R=ZZ/32003[a..c]; i2 : isLexIdeal lexIdeal(R,{1,3,4,3,1}) o2 = true i3 : isLexIdeal ideal(a^3-a^2*b) o3 = false i4 : isLexIdeal ideal(a^3,a^2*b) o4 = true i5 : isLexIdeal ideal(a^3,a^2*b,a^3-a^2*b) --not given as a monomial ideal but still a lex ideal o5 = true i6 : Q=R/ideal(a^3,b^3,a*c^2); i7 : isLexIdeal ideal(a^2*b,a^2*c) o7 = true i8 : isLexIdeal ideal(a^2*b,a*b^2) o8 = false