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Modular pairs, covering property and related results in posets. (English) Zbl 1117.06300

Summary: How should one define a modular pair in a general poset? This question was raised by Birkhoff sixty years ago. Not only answer we this open problem satisfactorily, but we, moreover, obtain interesting properties concerning covering property and 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, orthomodular posets are also characterized and an interesting open problem in this context is raised.

MSC:

06A06 Partial orders, general
06A11 Algebraic aspects of posets
06A12 Semilattices
06C05 Modular lattices, Desarguesian lattices
06C15 Complemented lattices, orthocomplemented lattices and posets
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