Stability for the zigzag submonoids

AbstractA notion of stability for the zigzag submonoids is introduced and called Z-stability. We prove that for the zigzag submonoids the Z-stability and the Z-freeness are equivalent, like the stability and the freeness for the ordinary submonoids. The class of Z-free zigzag submonoids is, however, not closed under intersection. We prove also that the property of being Z-stable is decidable for the regular zigzag submonoids