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Like this

sage: def PP(a,b,c): 
....:         return posets.ProductOfChains([a,b,c]).order_ideals_lattice() 
sage: PP(2,2,2)                                                                 
Finite lattice containing 20 element

Like this

sage: def PP(a,b,c): 
....:         return posets.ProductOfChains([a,b,c]).order_ideals_lattice() 
sage: PP(2,2,2)                                                                 
Finite lattice containing 20 element

for the full poset. Then you can take any principal order filter.

Like this

sage: def PP(a,b,c): 
....:         return posets.ProductOfChains([a,b,c]).order_ideals_lattice() 
sage: PP(2,2,2)                                                                 
Finite lattice containing 20 element

for the full poset. Then you can take any principal order filter.

EDIT Another way:

sage: P = PlanePartitions([4,3,2])                                                
sage: q = P.random_element()                                                      
sage: qt = q.to_tableau()                                                         
sage: Poset(([x.to_tableau() for x in P if x.to_tableau() <= qt] ,lambda x,y: x<=y))                                                                      
Finite poset containing 459 elements