Let [n] = {1, 2, . . . , n}, and let P On be the partial orderpreserving transformation semigroup on [n]. Let CP On = {α ∈ P On : (∀x, y ∈ domα), |xα − yα| ≤ |x − y|}.Then CP On is a subsemigroup of P On. In this paper, we characterize Green , s relations and the regularity of elements for CP On.

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By using Riccati transformation technique, we will establish some new oscillation criteria for the even order neutral delay differential equations z t n q t f x σ t 0, t ≥ t 0 , where n is even, z t x t p t x τ t , 0 ≤ p t ≤ p 0 < ∞, and q t ≥ 0. These oscillation criteria, at least in some sense, complement and improve those of Zafer 1998 andZhang et al. 2010 . An example is considered to illustrate the main results.

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