Let M be a cancellative and commutative (additive) monoid. The monoid M is atomic if every non-invertible element can be written as a sum of irreducible elements, which are also called atoms. Also, M satisfies the ascending chain condition on principal ideals (ACCP) if every increasing sequence of principal ideals (under inclusion) becomes constant from one point on. In the first part of this paper, we characterize torsion-free monoids that satisfy the ACCP as those torsion-free monoids whose submonoids are all atomic. A submonoid of the nonnegative cone of a totally ordered abelian group is often called a positive monoid. Every positive monoid is clearly torsion-free. In the second part of this paper, we study the atomic structure of certain classes of positive monoids.

令M为取消且可交换（可加）幺半群。如果每个不可逆元素都可以写为不可约元素（也称为原子）的和，则幺半群M是原子的。另外，米如果主理想（包含在内）的每个递增序列从一点开始变得恒定，则满足主理想的升序链条件（ACCP）。在本文的第一部分，我们将满足 ACCP 的无扭转幺半群表征为那些子幺半群都是原子的无扭转幺半群。全序阿贝尔群的非负圆锥的子幺半群通常称为正幺半群。每个正幺半群显然都是无扭转的。在本文的第二部分，我们研究某些类正幺半群的原子结构。