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Is the product of two objectos $a,b$ in a preorder category their infimum $a\wedge b$? I can't assume that their infimum exists, can I?

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The notation $a \wedge b$ denotes the infimum, actually. And the product of $a$ and $b$ exists if and only if the infimum exists. Dually, the coproduct exists if and only if the supremum exists. – John Myers Jan 7 at 16:32
@JohnM: thanks for the comment. – Barbra. Jan 7 at 16:45
@JohnMyers I edited the question, so now it looks correct – magma Jan 9 at 3:48

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Yes, the product of $a$ and $b$ is $a\land b$, if it exists; however, this is the infimum, not the supremum. As you suspect, it need not exist for arbitrary $a$ and $b$.

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Thanks for clarifying me. – Barbra. Jan 7 at 16:45
@Barbra: You’re welcome. – Brian M. Scott Jan 7 at 16:49

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