Free distributive lattice with any number of generators is finite. For example with 3 generator the lattice will have 20 elements.
Is there other examples of free objects that are finite and have at least ten elements?
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The free abelian-group-of-exponent-two on a finite number of generators is finite. – Mariano Suárez-Álvarez Jul 22 '15 at 07:58
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what do you mean by trivial here? As the question stands it highly unclear what you are looking for. – Ittay Weiss Jul 22 '15 at 08:03
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I changed the question a little. By "trivial" I meant something that obviously have only few elements. – Jori Mäntysalo Jul 22 '15 at 09:39
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An example: any finitely generated free idempotent semigroup is finite.
J.-E. Pin
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So, if generators are a, b and c, then for example abcabc=(abc)(abc)=abc, and the free object will contain strings without those kind of copied parts. Thank you! – Jori Mäntysalo Jul 22 '15 at 10:17
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...Or maybe not. See http://math.stackexchange.com/questions/3348/the-longest-string-of-none-consecutive-repeated-pattern . It seems to say that for $n \ge 3$ the semigroup will be infinite. – Jori Mäntysalo Jul 22 '15 at 19:29
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The free idempotent semigroup satisfies $x^2 = x$ for each element $x$. And I confirm it is finite if it is finitely generated. See Green, J.A., Rees, D.: On semigroups in which $x^r = x$. Math. Proc. Camb. Phil. Soc. 48, 35–40 (1952) – J.-E. Pin Jul 23 '15 at 08:32
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I made a new question of this: http://math.stackexchange.com/questions/1371255/free-idempotent-semigroup-with-3-generators – Jori Mäntysalo Jul 23 '15 at 12:39
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One notion you might be interested in is that of locally finite varieties; they satisfy an apparently stronger finiteness requirement, namely: Every finitely generated algebra in it is finite. Indeed, this is equivalent to having all finitely generated free algebra finite.
A prominent example is the class of Boolean algebras.
There is a great deal of study of this varieties, in particular with the question of which of them are decidable, meaning that there is an effective procedure to determine if a first-order formula is a consequence of the axioms. The book The structure of decidable locally finite varieties by McKenzie and Valeriote has deep results in this direction. I suggest that you take a look into its zbMATH review.
Pedro Sánchez Terraf
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One classical example, still unsolved in generality, is Burnside groups.
The free Burnside group $B(m,n)$ is defined to be the free group on $m$ generators with the relations $g^n=e$ for all $g\in B(m,n)$.
In general, it is unknown whether or not $B(m,n)$ is a finite group (though the answer is known in many cases). In fact, we already do not know whether $B(2,5)$ is finite.
Andrew Dudzik
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