PROYEKSI PADA ALJABAR MAX-PLUS; PROJECTION IN MAX-PLUS ALGEBRA

MUSTHOFA, Ari Suparwanto

2011 | Disertasi | PROGRAM STUDI S2 MATEMATIKA

Projection is idempotent linear mapping. In vector space V, the mapping P : V ® V is a projection if and only if V is direct sum of image P and kernel P . Since image P and kernel P are subspaces of V, then the existence of projection on the vector space is equivalent to the existence of two the subspaces V1 and V2 of V such that V is the direct sum of V1 and V2. For any subspace V1 of V, there exist B,C Î hom(V,V) such that V1 = im B and V1 = ker C. Consequently, the existence of projection on the vector space V is equivalent to the existence of B,C Î hom(V,V) such that V is direct sum of im B and ker C. In semimodule, kernel of an homomorphism is not subsemimodule. Consequently , if B: U ® X and C : X ® Y are semimodule homomorphisms, generalization is needed to determine the existence of projection on im B parallel to ker C. In semimodule, there is a projection on im B parallel to ker C if and only if for each x Î X, there exist a unique z Î im B, such that C(x) = C(z). Especially in max ? m´n semimodule, this condition is equivalent to the existence of K Î hom(X,U) and L Î hom (Y,X) such that C = CBK and B = LCB. The existence of K and L can be determined by getting the maximal element of { K / CBK ? C } and { L/ LCB ? B } from residuation of matrices over idempotent semiring. If the maximal elements are satisfy with C = CBK and B = LCB then P = BK = LC is a projection on image B parallel to kernel C.

Kata Kunci : Projection, Semiring , Semimodule