On Gröbner Basis and Cohen-Macaulay Property of Closed Path Polyominoes

In this paper we introduce some monomial orders for the class of closed path polyominoes and prove that set generators polyomino ideal attached to a forms reduced Gröbner basis with respect these orders. It is known containing an $L$-configuration or ladder at least three steps, equivalently having no zig-zag walks, prime. As consequence, obtain coordinate ring walks normal Cohen-Macaulay domain.