I asked this question on math stack exchange, but it didn't get any responses. So, I am asking it here. Suppose we are working in the signature of a single binary operation $*$. We are given a finite set $S$ of identities in that signature, and a single identity $E$ which is fixed in advance. Consider the decision problem of figuring out whether the set $S$ implies $E$. Is it the case that that problem is algorithmically decidable only in the trivial case of $E$ being a reflexive identity of the form $t=t$? Or, are there other identities for which the problem is algorithmically decidable? If there are, can someone give me an example of such an identity?
$\begingroup$
$\endgroup$
5
-
1$\begingroup$ The order of quantifiers is a bit unclear (to me). Are you asking whether $\exists E. \mathrm{computable}(\lambda S.S\Rightarrow E)$? $\endgroup$Daniel Wagner– Daniel Wagner2025-01-18 15:33:32 +00:00Commented Jan 18 at 15:33
-
1$\begingroup$ See also mathworld.wolfram.com/BirkhoffsTheorem.html $\endgroup$Gerald Edgar– Gerald Edgar2025-01-18 15:51:22 +00:00Commented Jan 18 at 15:51
-
1$\begingroup$ I like this theorem (Jacobson): If a ring satisfies $(\forall x) x^3=x$, then it also satisfies $(\forall x , y) xy=yx$. $\endgroup$Gerald Edgar– Gerald Edgar2025-01-18 16:01:52 +00:00Commented Jan 18 at 16:01
-
1$\begingroup$ @DanielWagner I am asking whether there is a specific identity, like maybe the associative identity or the commutative identity, such that the decision problem for that identity is decidable, aside from identities like $x=x$ $\endgroup$user107952– user1079522025-01-18 23:54:16 +00:00Commented Jan 18 at 23:54
-
$\begingroup$ As a test case, consider the identity $x=y$. This is implied by $S=\{x=x*y, x*y=y\}$, but it is not implied by $S_1=\{x=x*y\}$ or $S_2=\{x*y=y\}$. My guess is that for any nontrivial identity the problem is undecidable. $\endgroup$user551651– user5516512025-01-23 15:15:15 +00:00Commented Jan 23 at 15:15
Add a comment
|