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## Comments

The thing about your question is that we've been using \(\leq\) as the symbol for total orders, partial orders, and preorders. We also haven't really been using <, so I'm not completely sure what your question is. The thing you mentioned about the cycle implying that 11=12=22=21=11 is definitely true for partial orders, where all cycles must be trivial. The only difference in the definitions of partial order and preorder is that a preorder allows non-trivial cycles.

`The thing about your question is that we've been using \\(\leq\\) as the symbol for total orders, partial orders, and preorders. We also haven't really been using <, so I'm not completely sure what your question is. The thing you mentioned about the cycle implying that 11=12=22=21=11 is definitely true for partial orders, where all cycles must be trivial. The only difference in the definitions of partial order and preorder is that a preorder allows non-trivial cycles.`

Yes, as #Joseph suggested here the question goes about preorders, and the particular cycle in the example 1.33. I can readily imagine systems, which are strictly bigger from each other, actually example 1.2 and other examples show systems (partitions) which are partitioned by < relation ((+, *, #) < ((+, *), #). But all such systems form DAGs (directed acyclic graphs). In example 1.33 we were presented with a cycle, and my imagination only allows cycles if all systems forming a cycle are strictly equal. So I wondered whether there exists an (practical) example of such a cyclic preorder where while 1 < 2, 2 < 3, but 3 < 1?

`Yes, as #Joseph suggested here the question goes about preorders, and the particular cycle in the example 1.33. I can readily imagine systems, which are strictly bigger from each other, actually example 1.2 and other examples show systems (partitions) which are partitioned by < relation ((+, *, #) < ((+, *), #). But all such systems form DAGs (directed acyclic graphs). In example 1.33 we were presented with a cycle, and my imagination only allows cycles if all systems forming a cycle are strictly equal. So I wondered whether there exists an (practical) example of such a cyclic preorder where while 1 < 2, 2 < 3, but 3 < 1?`

Is this such an example with 6 elements, where 1 < 2, 2 < 3, 3 < 4, 4 < 5, 5 < 6, 6 < 1, or did I misunderstand?

`> I wondered whether there exists an (practical) example of such a cyclic preorder where while 1 < 2, 2 < 3, but 3 < 1? Is this such an example with 6 elements, where 1 < 2, 2 < 3, 3 < 4, 4 < 5, 5 < 6, 6 < 1, or did I misunderstand? ![compost cycle](https://user-images.githubusercontent.com/117439/38198435-767d2560-3652-11e8-978f-93fbd2ee8ae0.png)`

Generalizing the point made in #3 . If we use preorders to model a cyclical process in time there are many examples. e.g.

`Generalizing the point made in #3 . If we use preorders to model a cyclical process in time there are many examples. e.g. ![four stroke engine](https://upload.wikimedia.org/wikipedia/commons/d/dc/4StrokeEngine_Ortho_3D_Small.gif)`

@Bob, thanks for #3! This is quite a vivid example, indeed this is a cycle, and each time adding additional processing/energy we get something "bigger". @Fredrick, thanks for #4, that's also a cool remark about time, now I'm starting to understand all these better!

`@Bob, thanks for #3! This is quite a vivid example, indeed this is a cycle, and each time adding additional processing/energy we get something "bigger". @Fredrick, thanks for #4, that's also a cool remark about time, now I'm starting to understand all these better!`

@Igor Kulieshov: Just a heads up,

`#`

at the start of a line creates a headline. This forum uses a markdown variant for comments. @ (similar to twitter and others) is the way to reference other people (though it doesn't understand spaces in names, unfortunately).## h1

## h2

`@Igor Kulieshov: Just a heads up, `#` at the start of a line creates a headline. This forum uses a markdown variant for comments. @ (similar to twitter and others) is the way to reference other people (though it doesn't understand spaces in names, unfortunately). # h1 ## h2`

Thanks for the tip, @Jared.

`Thanks for the tip, @Jared.`

You can mention a user via a markdown link on their username on their forum comment, like Igor Kulieshov. I have no idea if they get a notification, though.

You can mention their comment the same way using the link on the time on the comment,, like Igor's comment 7.

`You can mention a user via a markdown link on their username on their forum comment, like [Igor Kulieshov](https://forum.azimuthproject.org/profile/1923/Igor%20Kulieshov). I have no idea if they get a notification, though. You can mention their comment the same way using the link on the time on the comment,, like [Igor's comment 7](https://forum.azimuthproject.org/discussion/comment/16474/#Comment_16474).`

Thanks Bob Haugen, it is a helpful suggestion!

`Thanks [Bob Haugen](https://forum.azimuthproject.org/profile/1711/Bob%20Haugen), it is a [helpful suggestion](https://forum.azimuthproject.org/discussion/comment/16488/#Comment_16488)!`