以下是poset [{a, b, c, d, e}, ≤]的Hasse图
该poset是
(A)不是格子
(B)一个格但不是一个分配格
(C)一个分配格而不是一个布尔代数
(D)布尔代数答案：(乙)
解释：
It is a lattice but not a distributive lattice.
Table for Join Operation of above Hesse diagram
V |a b c d e
________________
a |a a a a a
b |a b a a b
c |a a c a c
d |a a a d d
e |a b c d e
Table for Meet Operation of above Hesse diagram
^ |a b c d e
_______________
a |a b c d e
b |b b e e e
c |c e c e e
d |d e e d e
e |e e e e e
Therefore for any two element p, q in the lattice (A,<=)
p <= p V q ; p^q <= p
This satisfies for all element (a,b,c,d,e).
which has 'a' as unique least upper bound and 'e' as unique
greatest lower bound.
The given lattice doesn't obey distributive law, so it is
not distributive lattice,
Note that for b,c,d we have distributive law
b^(cVd) = (b^c) V (b^d). From the diagram / tables given above
we can verify as follows,
(i) L.H.S. = b ^ (c V d) = b ^ a = b
(ii) R.H.S. = (b^c) V (b^d) = e v e = e
b != e which contradict the distributive law.
Hence it is not distributive lattice.
so, option (B) is correct. 这个问题的测验