Determinantal structure

From Apstheory

[edit] Definition

[edit] For an IAPS of groups

A determinantal structure on an IAPS of groups is defined as a homomorphism from that IAPS to a constant IAPS for an Abelian group. We call the determinantal structure standard if it has the additional property that the Abelian group is the Abelianization of the ground member and that the map restricted to the ground member is simply the canonical map to the Abelianization.

If the IAPS of groups is Template:Fillin, then there exists at most one determinantal structure.

A natural example is the determinantal structure on the GL IAPS.

[edit] For an IAPS of monoids

A determinantal structure on an IAPS of monoids is defined as a homomorphism from that IAPS to a constant IAPS over an Abelian monoid. We call the determinantal structure standard if it has the additiona property that the Abelian monoid is the Abelianization of the ground member and that the map restructed to the ground member is simple the canonical map to the Abelianization.

An example is the determinantal structure on the matrix IAPS.