Modular Pairs, Covering Property and Related Results in Posets

How should one define a modular pair in a general poset? queried Birkhoff, sixty years ago. Not only we answer this open problem satisfactorily but obtain interesting properties concerning covering property, exchange property in a poset with zero. In this context a few counter examples are also supplied. This general study has led us, as an offshoot, to thirteen characterizations of covering property in a lattice with zero. Del-relation and perspectivity are characterized in posets. The study of atom spaces is extended to posets. Statischness in atomistic posets is characterized. Further, ortho-modular posets are also characterized and an interesting open problem in this context is raised.