  Macaulay posets are posets in which an analog of the Kruskal-Katona Theorem
holds. Macaulay rings (also called Macaulay-Lex rings) are rings in which an
analog of Macaulay's Theorem for lex ideals holds. The study of both of these
objects started with Macaulay almost a century ago. Since then, these two
branches have developed separately over the past century, with the last link
being the Clements-Lindstr\"om Theorem.
  For every ring that is the quotient of a polynomial ring by a homogeneous
ideal we define the poset of monomials. Under certain conditions, we prove a
Macaulay Correspondence Theorem, a ring is Macaulay if and only if its poset of
monomials is Macaulay. Furthermore, the tensor product of rings corresponds to
the Cartesian product of the posets of monomials. This allows us to transfer
results between rings and posets. By using this translation, we give several
answers to a problem posed by Mermin and Peeva, a positive answer to Hoefel's
question about applying Macaulay poset theory to ring theory, and deduce
several other results in algebra and combinatorics.
  A new proof of the Mermin-Murai Theorem on colored square free rings is
presented by using star posets. We extend the Mermin-Murai Theorem to rings
that are not square free. Using a result from Mermin and Peeva we give an
answer to a question posed by Bezrukov and Leck. Some results of Chong also
give answers to the Bezrukov-Leck problem. All of these results have a common
feature. They involve the tensor product of rings whose Hasse graphs of the
poset of monomials are trees. We call such rings, tree rings. We give a
classification of Macaulay rings that are the tensor product of a tree ring.
Finally, we show that there are Macaulay rings that are not the tensor product
of tree rings, and present the first examples of Macaulay rings that are not
quotients by a monomial ideal and not quotients by a toric ideal.
