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# Def/Lax basis

Definition of Lax basis: Suppose that $[a,b]$ and $[c,d]$ are lax vectors in $\ZZ^2$. We say that the unordered pair $\{ [a,b], [c,d] \}$ is a lax basis of $\ZZ^2$ if the following condition holds:

• For all $u,v \in \ZZ^2$, there exist $x,y \in \ZZ^2$, such that:

$$ax + by = u,$$ $$cx + dy = v.$$

By State/Systems of two linear Diophantine equations, this is equivalent to the following condition:

• $ad - bc = \pm 1$.

From this equivalent condition, it is clear that the notion of lax basis is well-defined; it depends only upon the unordered pair of lax vectors $[a,b]$ and $[c,d]$, and not upon a choice of representatives $\pm (a,b)$ and $\pm (c,d)$ in $\ZZ^2$.

## Logical Connections

This definition logically relies on the following definitions and statements: Def/Lax vector, Def/Unordered pair, State/Systems of two linear Diophantine equations

The following statements and definitions logically rely on the material of this page: Def/Domain topograph, Def/Lax superbasis, State/Arithmetic progression rule for binary quadratic forms, State/Every lax basis is contained in exactly two lax superbases, State/Every lax vector in a lax basis is primitive, and State/Every primitive lax vector belongs to a lax basis

To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Binary quadratic forms