If I understand what you are getting at, No, transitivity doesn't hold.
For example, if we have the following vectors of numbers:
I = 1,2,3,3,4,5
II = 2,3,4,4,5,6
III = 3,4,5,5,6,7
By t-test, the mean of I is not significantly different than the mean of II. Likewise the mean of II is not significantly different than the mean of III. Yet the mean of I is significantly different than the mean of III.
We could summarize these results in a compact letter display, where the groups sharing a letter are not significantly different:
I a
II ab
III bc
I think transitivity does not apply to a statistical level. You'll see, in statistic, it's all about variance and distribution, not only mathematical differences. So, Salvatore has a very good point. That's why we talk about statistical significance differences and not mathematical differences, because of a statistical level, variance matters.
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