Timeline for Which algebraic theories have the property that $\mid$ is antisymmetric for all free algebras?
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11 events
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Jul 16, 2014 at 13:30 | vote | accept | goblin GONE | ||
| Jul 14, 2014 at 14:46 | answer | added | Benjamin Steinberg | timeline score: 4 | |
| Jul 14, 2014 at 14:26 | comment | added | Benjamin Steinberg | In semigroup theory this preorder is called the $\mathcal J$-order. A semigroup is called $\mathcal J$-trivial if it is a partial order. There are a number of natural varieties of semigroups and monoids where all free algebras are J-trivial. Note J-triviality is given by finitely many quasiidentities. For finitary universal algebras you may need infinitely many. | |
| Jul 14, 2014 at 13:51 | comment | added | goblin GONE | @BenjaminSteinberg, yes; it seems I misunderstood you. | |
| Jul 14, 2014 at 12:45 | comment | added | Benjamin Steinberg | But the theory is the theory of monoids and semigroups. In your question you only ask about the relation on free objects. | |
| Jul 14, 2014 at 7:19 | comment | added | goblin GONE | @BenjaminSteinberg, or rather, for free monoids / semigroups etc. Some monoids happen to be groups, after all. | |
| Jul 14, 2014 at 2:00 | comment | added | Benjamin Steinberg | This relation is antisymmetric for monoids and semigroups without abelian and for many natural subvarieties. | |
| Jul 13, 2014 at 20:32 | comment | added | Gerhard Paseman | I also suspect that such algebras will yield a normal form, or provide a small "unnormalizable" core. You might look at Knuth-Bendix to see if someone has come to a similar conclusion. Gerhard "Or Tweak The Relational Properties" Paseman, 2014.07.13 | |
| Jul 13, 2014 at 20:24 | comment | added | Gerhard Paseman | In model theory, one has some notion (algebraic with respect to parameters a) that is related to this. I don't know if there is a standard name for the preorder. For finite algebras, there is a classification of such algebras for which (I suspect) the relation is total. You might find Hobby and McKenzie's text on tame congruence theory of interest, especially the foundational part. Your notion can be extended by replacing = by an arbitrary congruence of X, and may have been considered for finite X by Hobby and McKenzie. Gerhard "Doing This With Fuzzy Memory" Paseman, 2014.07.13 | |
| Jul 13, 2014 at 19:03 | history | asked | goblin GONE | CC BY-SA 3.0 |