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By Richard Bellman (ed.)

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G ( x i , . . , x n , 2 / i , . . ,ym). }) = cr(x) A 0 & V{(ei(xi) a i + . . + e fc (x fc ) a J(:r) | {xi)ai G 5 , ei = ± 1 , i = 1 , .

Vm} and the operation set {fl, U, * i , . . , *&}. , Cm be classes of fuzzy subsets of X that are closed under projections. ,fim) holds for all fuzzy sets fii G C;, i = 1, 2 , . . , ra, w/iere REL is any one of the three relations Proof. We consider only the case in which REL is C, since the case of 2 follows by symmetry and then the case of = follows by combining C and D . ,fjLm)(x). . , /i m )(x). ,£H(/x m )(r))(x) = = = £(*(Mi), •.. B but Let A be an (A, V, * i , . . , *fc)-algebra.

Let x be an L-subset of G such that x ^ / i . Let a be the L-subset of G defined by V y G G, o~{y) = V{(ei(xi) a]L + . . , et = ± 1, z = 1 , . . , n; n G N}. 27ien cr = ( x ) . Proof. 8, e;(z;) a . C (x) for i — 1 , . . , n. Thus cr C (x) since e i ( x i ) a i + . . 7. To show that (x) £ a, it suffices to show that a G L(/J,) and x Q ^ Let x G G and x(#) = a. Then clearly xa{x) < a(x) since x a = e(xa) with e = 1 and n = 1. Hence x £ 0\ Let x, y E G. Then

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