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Is it okay to ask for help in clarifying a thought? Here's the particular thought I want help clarifying.

It seems that conservative extensions can alter the notion of structure homomorphism (but not isomorphism). In particular, adding new relations, even when they can be defined in terms of the old relations, may result in a stricter notion of homomorphism.

e.g. Consider a poset $(X,\leq)$ that is a meet-semilattice. We can conservatively extend this poset to produce a new structure $(X,\leq,\wedge)$ by defining that $x \wedge y = z$ if and only if $z$ is the least upper bound of $\{x,y\}$. This is extension by definition.

Now every (structure) homomorphism $(X,\leq,\wedge) \rightarrow (Y,\leq,\wedge)$ is necessarily a structure homomorphism $(X,\leq) \rightarrow (Y,\leq)$. But the converse may fail. That's because, in more cause speak, "A subposet of a lattice that just happens itself to be a lattice may fail to be a sublattice."

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  • $\begingroup$ What is the question you want to ask (what do you want clarified)? After the cut you gave a bunch of statements. Is there anything about it that you do not understand fully? As written I do not see how the collection of statements under the cut constitute a "question". $\endgroup$
    – Willie Wong
    Commented Feb 12, 2013 at 14:01
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    $\begingroup$ However, it can be a perfectly reasonable question if you were to add something like "Is it generally true that conservative extensions alter the notion of structure homomorphisms?" or "How do conservative extensions interact with structure homomorphisms?" or "Is my example below with the poset correct?" As written it is entirely unclear what parts of the "thought" requires clarification. $\endgroup$
    – Willie Wong
    Commented Feb 12, 2013 at 14:03

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Where this question has no definite answer, here is what I believe: If you have your thoughts well-written in there, you may surely ask a question in clarifying a thought. Make sure you put everything into it (or else the deadly close votes). Although, there are exceptions as well — and if you feel that your question is not totally a real question, or it is localised, use the Math SE chat. You should definitely not ask for definitions of a term, for example:

What is algebra?

The above also has exceptions. But that is when you don't just ask the question by itself: you post the background and your thought-process. If you really are confused, then you are welcome to ask on meta like this (or else, drop by the chat to confirm)!

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  • $\begingroup$ Missing a “not”. Thanks $\endgroup$
    – P.K.
    Commented Feb 12, 2013 at 17:17
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I think so, if you make sure you indicate clearly what you are looking for, or what needs clarification. I asked a question of this sort last week and nobody complained.

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