Macaulay–Lex rings

We present simple proofs of Macaulayʼs theorem and Clements–Lindströmʼs theorem. We generalize Shakinʼs theorem by proving that a stable ideal I of S is Macaulay–Lex if and only if I is a piecewise lexsegment ideal. We also study Macaulay–Lex ideals of the form , where 2⩽e1⩽⋯⩽en⩽∞ and ti