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Order Laws

Proposition: Let <= be a partial order on S. Then for every x, the following holds:

x is least (greatest) element of S w.r.t. <= =>

x is minimal (maximal) element of S w.r.t. <= ,

x is least (greatest) element of S w.r.t. <= =>

x is infimum (supremum) of S w.r.t. <= ,

x is lower (upper) bound of S w.r.t. <= /\  x in S =>

x is least (greatest) element of S w.r.t. <= .