Complete characterization of algebraic properties for non-simple polyominoes

Characterize completely, for non-simple polyominoes (finite collections of unit grid cells joined edge-to-edge that contain at least one hole), which instances have prime inner 2-minor ideal I_P, which have Cohen–Macaulay coordinate ring K[P]=S_P/I_P, and which have Gorenstein coordinate ring, thereby providing a full classification of primality, Cohen–Macaulayness, and Gorensteinness in the non-simple case.

Background

The paper studies binomial ideals arising from collections of cells and associated coordinate rings K[P], focusing on properties such as primality, Cohen–Macaulayness, and Gorensteinness. For simple polyominoes (no holes), these properties are well understood and K[P] is known to be a normal Cohen–Macaulay domain.

The authors highlight that attention has shifted to non-simple polyominoes, yet a full classification of the key algebraic properties remains unresolved. This motivates their investigation of specific non-prime classes (zig-zag collections and closed paths) as steps towards broader understanding.