Title: On mesoprimary decomposition of monoid congruences,
Abstract: We prove two main results concerning mesoprimary decomposition of monoid
congruences, as introduced by Kahle and Miller. First, we identify which
associated prime congruences appear in every mesoprimary decomposition, thereby
completing the theory of mesoprimary decomposition of monoid congruences as a
more faithful analog of primary decomposition. Second, we answer a question
posed by Kahle and Miller by characterizing which finite posets arise as the
set of associated prime congruences of monoid congruences.