Letterplace and co-letterplace ideals of posets

To a natural number n, a finite partially ordered set P and a poset ideal J in the poset Hom(P,[n]) of isotonian maps from P to the chain on n elements, we associate two monomial ideals, the letterplace ideal L(n,P;J) and the co-letterplace ideal L(P,n;J). These ideals give a unified understanding of a number of ideals studied in monomial ideal theory in recent years. By cutting down these ideals by regular sequences of variable differences we obtain: multichain ideals and generalized Hibi type ideals, initial ideals of determinantal ideals, strongly stable ideals, d-partite d-uniform ideals, Ferrers ideals, edge ideals of cointerval d-hypergraphs, and uniform face ideals.