Pablo @ 55: is it fair to say, then, that all elements of \\(A\\) must belong to a ~-connected and ~-closed subset of \\(A\\)?
I thought of a few possible examples of posets on the train this morning, that are maybe isomorphic to the ones already described but would represent interesting scientific applications.
1) level of detail in a worldview being generated by mobile agent performing information/feature fusion on a WSN (6.1.4 of Nakamura Et. Al.) \\(\leq\\) would be an operator comparing "level of detail" or "breadth of knowledge", and would be used to arbitrate conflicts in decisions drawn from different agents
2) the dependency graph of spacecraft failures, where \\(\leq\\) would indicate "leads to" or "increases the likelihood of"
3) circumstantial evidence in a set of experiments toward a conclusion in a life detection experiment (such as McKay et. al.), where \\(\leq\\) would indicate that a conclusion is "more likely to be caused by life"
Nakamura, Eduardo F., Antonio AF Loureiro, and Alejandro C. Frery. "Information fusion for wireless sensor networks: Methods, models, and classifications." ACM Computing Surveys (CSUR) 39.3 (2007): 9.
McKay, David S., et al. "Search for past life on Mars: possible relic biogenic activity in Martian meteorite ALH84001." Science 273.5277 (1996): 924-930.