This paper deals with su‰cient conditions on a poset in order to get it spectral. A motivating question is the following (p. 833 [LO76]): "If X is a height 1 poset such that for all x 0 y A X ; "x V "y and #x V #y are finite, is X spectral?" We obtain the some su‰cient conditions for such a poset X to be spectral. In particular, we prove that either if there is a finite subset F J X such that #F K Min X , or if diam X a 2, then the poset X is spectral.