Timeline for Free algebras from model theory perspective
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| May 15, 2022 at 21:33 | history | became hot network question | |||
| May 15, 2022 at 18:09 | comment | added | Benjamin Steinberg | The number of generators of a free semigroup or monoid is determined by the first order theory. The generators are the irreducible elements. The first order theory encodes arithmetic. | |
| May 15, 2022 at 16:43 | comment | added | Keith Kearnes | @arunpatel: regarding your request for a paper about free algebras in the variety of Jonsson-Tarski algebras, a good one is "Bouscaren, Elisabeth; Poizat, Bruno, Des belles paires aux beaux uples. J. Symbolic Logic 53 (1988), no. 2, 434-442". They show that the theory of the free algebras in this variety has QE, is stable, 1-based, and has the DOP. | |
| May 15, 2022 at 16:29 | answer | added | Keith Kearnes | timeline score: 7 | |
| May 15, 2022 at 16:23 | comment | added | Emil Jeřábek | Which, by the way, contradicts your claim about $\odot$. Your argument only works when $\odot$ is power-associative (and you need all powers of $a$ to be distinct, not just the second power). | |
| May 15, 2022 at 16:21 | comment | added | Emil Jeřábek | When $\mathbb V$ is the variety of all algebras of a given signature, or more generally, a variety axiomatized by “commutativity” equations postulating that some of the functions are symmetric wrt specific permutation groups, the first-order theory of $F_S$ has been described and studied by Malcev, Axiomatizable classes of locally free algebras of several types (mi.mathnet.ru/eng/smj/v3/i5/p729, doi.org/10.1016/S0049-237X(08)70560-3). These theories are all stable (though this is not in Malcev’s paper, which predates the definition of stability). | |
| May 15, 2022 at 13:52 | comment | added | arunpatel | That sounds like exactly what I'm looking forward, thank you. I haven't heard of those algebras, do you have any papers in mind? | |
| May 15, 2022 at 13:39 | comment | added | YCor | The variety of Jonsson-Tarski algebras (set $X$ endowed with bijective map $X^2\to X$) has been studied from the model-theoretic point of view too (model theorists call them "set with pairing function, if I remember correctly). | |
| May 15, 2022 at 13:36 | history | edited | YCor | CC BY-SA 4.0 |
formatting, added tag
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| May 15, 2022 at 13:31 | history | edited | arunpatel | CC BY-SA 4.0 |
edited title
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| May 15, 2022 at 13:30 | history | edited | arunpatel | CC BY-SA 4.0 |
added 10 characters in body
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| S May 15, 2022 at 13:29 | review | First questions | |||
| May 15, 2022 at 13:56 | |||||
| S May 15, 2022 at 13:29 | history | asked | arunpatel | CC BY-SA 4.0 |