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The join-meet ideal was introduced by Takayuki Hibi in 1987. It is binomialideals that are defined by finite lattices. We study the join-meet ideal ofnon-distributive finite lattices that do not always satisfy modular. Inparticular, we work on the case of Crystal lattice which is one of them. Inthis paper, we give the computational result of a total of each of the 1-th and2-th of grade Betti numbers of the join-meet ideal of a special Crystallattice. The important point about this result is that it is characterized bythe number of combinations of special generators. Moreover, we can considerfrom this result that a compatible monomial order suggests a connection tosomething good.

Total of each of the 1-th and 2-th Betti numbers of the join-meet ideal of Special Crystal Lattice