I have a RNAi knock-down experiment using a cell line, and did 4 (dye-swapped) arrays comparing knock-down and control. My question is, if I repeat the knock-down experiment with the same cell line and same siRNA molecule (and I do not have reason to believe the results should be different from the first experiment), do 4 more dye-swapped arrays, are the two sets of arrays technical or biological replicates? Which is the same as: are the knock-down experiments technical or biological replicates? I know if I changed the cell line or the siRNA molecule it would be biological replicate, but keeping both cell line and siRNA, and all other conditions as well (eg. temperature, time until RNA extraction, etc), I do not know if it technical or biological replication.
Your second experiment, with the same cell line and siRNA would be a biological replicate. As you said, you would not expect to see strong differences between your first and second biological replicates, under the assumption that the cells in both replicates are in essentially the same 'state.'
If you were to conduct another experiment with a different cell line or siRNA, that would not be a replicate at all, based on your description.
I'm not totally clear on the nature of your 4 arrays. If they are redundant (ie. they have the same probes on them) then those are technical replicates. As you can see, it is entirely possible to have 'nested' sets of replicates, in this case it looks like some technical replicates nested within biological replicates.
For what it's worth, that sounds like a sound study design to me. As usual, larger sample sizes are better but we're always constrained by budget, time etc.
My question is, if I repeat the knock-down experiment with the same cell line and same siRNA molecule (and I do not have reason to believe the results should be different from the first experiment), [...] keeping both cell line and siRNA, and all other conditions as well (eg. temperature, time until RNA extraction, etc), I do not know if it technical or biological replication.
If you repeat the experiment, and keep everything the same, it is a biological replicate. This is because sources of variation are everywhere, and even though you think you are keeping everything the same, you can't possibly keep everything the same between both trials. Consider that you don't walk down a hallway the same way twice. Even though you might be working in a controlled environment there are fluctuations: the air quality can differ (maybe the gardeners are mowing the lawn today, maybe it's a high ozone day in your area), one of your reagents might be from a different batch, some of your tubes have been taken out of the freezer one more cycle, maybe someone left your pipet tip box open over night but closed it before you came in. The efficiency of a knockdown is not the same from trial to trail. Even though it's the same cell line, the gene expression state will fluctuate and show variation in time. Sitting in your office at noon on Tuesday and Wednesday, will your blood pressure be identical at both time points? If not, is day of the week the causative agent? Variation is everywhere, how you isolate it, how you aliquot it determines the level (i.e. technical vs. biological) of replication. You're friend is wrong that splitting a cell line is the same as splitting an RNA extraction. There is more variation between two aliquots of living cells, than between a split aliquot of extracted RNA molecules. Since you began each experiment with an independent entity of cells, you have a biological replicate.
Dan and seidel's answers are great. Here a few more general thoughts on the topic.
To a certain degree the definition or description of replicates may be relative and depend on the question you are asking. However, one way to think of this issue is as follows. A purpose of replicates is to estimate or account for the variability in a measurement or observation. These estimates speak to the reproducibility and ultimately the precision of an experiment. It doesn't really matter what you call your replicates, the key is the think about the source of the variability you seek to estimate by performing your replicates. Vigilantly thinking about the nature and sources of variability is a critical step towards developing sound experimental designs. Also it is never too early to think about what statistical approaches/tests you are going apply in evaluating your replicates. Form hypotheses about the sources of variability and their relative magnitude and contribution to your overall variability and seek to test those hypotheses.
Ask yourself, is the variability I expect to encounter across my replicates related to a technical or biological process? For example, in the experiment you describe you are using multiple microarrays. The manufacturing of these arrays is not an exact process. Replicates where the only variable changing is the microarray slide being used are going to be considered technical in nature. Similarly, scanning of the microarrays is not perfectly reproducible and introduces variance of a technical nature. Replicates that seek to estimate the variability associated with anything related to manufacturing, instrumentation (e.g. variability in pipetting volumes when you add your siRNA reagent), reagent batches, etc. related to the 'technological' aspects of your method can likely be classed as 'technical replicates'. You may have some grey areas where a biological process (e.g. enzymes) are involved in the creation of a reagent. Experiments that involve tissue culture, model organisms, etc. are going to have both technical and biological sources of variation. For example, RNA expression patterns measured from the cells in your cultures will vary according to cell cycle of cells in the culture, mutations acquired during culture, and possibly many other biological processes related to cell growth, etc. Replicates that seek to measure variability associated with such factors can likely be considered 'biological replicates'. If you seek to demonstrate that an observation is general to any cell type, you might consider repeating the experiment in different cell lines to also be 'biological replicates'. However, as mentioned in the comments, this may introduce so many sources of biological variability that it is almost of a different experiment.
Finally, as has already been noted, a replicate may involve an overlap of both technical and biological sources of variability. Biological replicates in particular are usually influenced by both technical and biological processes. The generic term 'experimental replicate' may be used to describe a replicate of the entire experiment comprising a complicated mix of multiple sources of variability.