On ideal monotone sequences

In this paper, we introduce the notion of an ideal monotone sequence of real numbers,related to a given ideal, generalizing an earlier concept known as statistically monotonesequence.We present some relationships between the ideal monotonicity of a given sequence andits subsequences for certain classes of ideals. The results concerning ideal monotonicitypresented here are also closely related to earlier results regarding ideal convergence andideal boundedness. We use Lebesgue measure and Baire category as gauges of size, whenstudying classes of ideal monotone subsequences of a real valued sequence.