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Stream: learning: questions

Topic: In a quantale, $$ a * \_ $$ is monotone

fosco (Aug 19 2022 at 20:12):

To my shame, there's a seemingly simple fact about quantales that I don't see how to prove: let $a,b,c$ be three elements of a quantale $Q$, and assume $b \le c$; then $a * b \le a * c$ (and similarly multiplying with $a$ on the other side).

This looks simple, right? So then why I can't see how to prove it, starting with the definition of quantale, that apparently does not require it (see for example the definition given in Kimmo Rosenthal's book: image.png ?

Mike Shulman (Aug 19 2022 at 20:15):

Note that $b\le c$ if and only if $c = b\vee c$. Thus, if $b\le c$ then by the axiom given, $a \ast c = a\ast (b\vee c) = (a\ast b) \vee (a \ast c)$, which is equivalent to $a \ast b \le a \ast c$.

fosco (Aug 19 2022 at 20:16):

ah, of course!