Timeline for Every homomorphism between (rational) Puiseux monoids is multiplication by a non-negative rational
Current License: CC BY-SA 4.0
8 events
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| Feb 13 at 7:27 | comment | added | Salvo Tringali | Yes, I agree. I just wanted to emphasize that the result in the OP is not exactly the same as Corollary 4 in Sasaki and Tamura's paper. It is rather that the former is a straightforward corollary of the latter. | |
| Feb 13 at 4:15 | comment | added | Benjamin Steinberg | @SalvoTringali, yes. This is easy to see though since any two elements have a common multiple and so if one element goes to 0 they all do. | |
| Feb 13 at 3:56 | comment | added | Salvo Tringali | So, to recover the result in the OP from Sasaki and Tamura's corollary, I think one should first argue (as trivial as it may be) that a monoid hom $f \colon H \to K$ between Puiseux monoids restricts to a sgrp hom between the "positive cones" of $H$ and $K$, unless $f$ is identically zero. | |
| Feb 13 at 3:47 | comment | added | Salvo Tringali | I think that, by the term "positive rational semigroup", Sasaki and Tamura mean a (non-empty) subsemigroup of the positive rational numbers under addition. Corollary 4 in their paper says that any homomorphism between positive rational semigroups is in fact an isomorphism (of the form $x \mapsto rx$, where $r$ is a positive rational number). This is not the case with Puiseux monoids, but only because it is missing the zero homomorphism. | |
| Feb 13 at 3:41 | comment | added | Benjamin Steinberg | I am confused. I think what they mean by positive rational semigroup is exactly puiseux monoids (except without the identity) and Cor 4 says every homomorphism between them is given by multiplication by a rational | |
| Feb 13 at 3:30 | vote | accept | Salvo Tringali | ||
| Feb 13 at 3:29 | comment | added | Salvo Tringali | Thanks! Corollary 4 in Sasaki and Tamura's paper does indeed imply the result in the OP, after noticing that a monoid homomorphism between Puiseux monoids must either be identically zero or map positive rationals to positive rationals (and hence restrict, in the terminology of Sasaki and Tamura, to a semigroup homomorphism of positive rational semigroups). Before asking here, I had the term "rational semigroup" in mind but couldn't remember why... | |
| Feb 13 at 2:25 | history | answered | Benjamin Steinberg | CC BY-SA 4.0 |