Working with partial order relations in discrete math.

Let S = {0,1} and consider the partial order relation R defined on S X S X S as follows: for all ordered triples (a, b, c) and (d, e, f) in S X S X S.
( a, b, c ) R ( d, e, f ) <-> a &#8804; d, b &#8804; e, c &#8804; f,
where &#8804; denotes the usual "less than or equal to" relation for real numbers. Do the maximal, greatest, minimal and least elements exist? If so, which are they?

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... the greatest element. We know a maximal element is the greatest one in a partial order chain. This element, say (a,b,c) has to be (1,1,1). If not, then a<1 or b<1 or c<1. In each case we have (a,b,c)R(1,1,1) and ...