# isLexModule -- whether a monomial module over an exterior algebra is lex

## Synopsis

• Usage:
isLexModule M
• Inputs:
• M, a monomial module over an exterior algebra
• Outputs:
• , whether the module M is lex

## Description

A monomial module M is lex if for all monomials $u,v$ of F of the same degree with $v\in M$ and $u>v$ (> lex order) then $u\in M$. An equivalent definition of a lex submodule is the following one: a monomial submodule $M=\oplus_{i=1}^{r}{I_ig_i}$ of F is lex if $I_i$ is a lex ideal of E for each $i,$ and $(e_1,\dots, e_n)^{\rho_i + f_i - f_{i-1}} \subseteq I_{i-1}$ for $i = 2, \dots, r$ with $\rho_i = \mathrm{indeg}\ I_i.$

Example:

 i1 : E = QQ[e_1..e_4, SkewCommutative => true] o1 = E o1 : PolynomialRing, 4 skew commutative variables i2 : F=E^{0,0} 2 o2 = E o2 : E-module, free i3 : I_1=ideal(e_1*e_2,e_1*e_3,e_2*e_3) o3 = ideal (e e , e e , e e ) 1 2 1 3 2 3 o3 : Ideal of E i4 : I_2=ideal(e_1*e_2,e_1*e_3) o4 = ideal (e e , e e ) 1 2 1 3 o4 : Ideal of E i5 : M=createModule({I_1,I_2},F) o5 = image | e_2e_3 e_1e_3 e_1e_2 0 0 | | 0 0 0 e_1e_3 e_1e_2 | 2 o5 : E-module, submodule of E i6 : Malex=almostLexModule M o6 = image | e_1e_4 e_1e_3 e_1e_2 e_2e_3e_4 0 0 | | 0 0 0 0 e_1e_3 e_1e_2 | 2 o6 : E-module, submodule of E i7 : isLexModule Malex o7 = false i8 : L=createModule({ideal(e_1*e_2,e_1*e_3*e_4),ideal(e_1*e_2*e_3*e_4)},F) o8 = image | e_1e_2 e_1e_3e_4 0 | | 0 0 e_1e_2e_3e_4 | 2 o8 : E-module, submodule of E i9 : isLexModule L o9 = true