Geodesic
The 3-Irreducible Partially Ordered Sets
Canadian journal of mathematics, Tome 29 (1977) no. 2, pp. 367-383

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The dimension [4] of a partially ordered set (poset) is the minimum number of linear orders whose intersection is the partial ordering of the poset. For a positive integer m, a poset is m-irreducible[10] if it has dimension m and removal of any element lowers its dimension. By the compactness property of finite dimension, every m-irreducible poset is finite and every poset of dimension ≧ m contains an m-irreducible subposet. 
Kelly, David. The 3-Irreducible Partially Ordered Sets. Canadian journal of mathematics, Tome 29 (1977) no. 2, pp. 367-383. doi: 10.4153/CJM-1977-040-3
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