There are many examples of nonfinitely based varieties. In a finite signature, is there an example of such variety with a known explicit set of identities?

1$\begingroup$ How explicit do you need? There are the identities satisfied by Murskii's groupoid on 3 elements., $\endgroup$ – The Masked Avenger Jan 16 '14 at 15:11

$\begingroup$ @Masked Avenger: This is what I just need. Give a reference please! $\endgroup$ – M. Shahryari Jan 16 '14 at 15:17

$\begingroup$ Sorry, my memory is too fuzzy. A web search will reveal it. The original monograph is in Russian about 1965 or maybe 1967. Many have written about it and similar algebras, including George McNulty and Mark Sapir. Using "Murskii finitely based" should get you started. $\endgroup$ – The Masked Avenger Jan 16 '14 at 15:25
You can look at "Bases for Equational Theories of Semigroups" by P Perkins, J Algebra 11, 298314 (1968). Theorem 2: the identities
$xyzw=xzyw$ and $yx^ky=xyx^{k2}yx$ for $k=2,3,\dots$
define a nonfinitely based variety of semigroups.

$\begingroup$ Thank you, I need this kind of examples to construct noneqautional noetherian relatively free algebras: Let $V$ be the variety of semigroups and $W$ be the subvariety defined by the identities just you introduced. Since the defining identities of $W$ are written using 4 variables, and $W$ is not finitely based, so $F_V(x, y, z, w)$ is not equational noetherian. An interesting example! $\endgroup$ – M. Shahryari Jan 16 '14 at 21:01
There are a few examples that are finitely generated.
(1) Let $L$ be Lyndon's groupoid given by the following multiplication table: \begin{array} [c]{cccccccc} L & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 4 & 0 & 4 & 5 & 6 & 0 & 0 & 0 \\ 5 & 0 & 5 & 5 & 5 & 0 & 0 & 0 \\ 6 & 0 & 6 & 6 & 6 & 0 & 0 & 0 \\ \end{array} Then the variety $\mathrm{var}\, L$ is nonfinitely based and an explicit basis is: \begin{align} (xx)y = x(yz) = zz, \quad (\cdots((xy_1) y_2) \cdots) y_k = ((\cdots((xy_1) y_2) \cdots) y_k) y_1, \\ ((\cdots(x_1 x_2) \cdots )x_k )x_1 = zz, \quad k=1,2,\ldots \end{align}
(2) Let $A_2$ be the 0simple semigroup $$ \langle a,b \mid a^2=aba=a,\ bab=b,\ b^2=0\rangle $$ of order five and let $\mathbb{Z}_n$ be the cyclic group of order $n$. Then for each $n \geq 2$, the variety $\mathrm{var} \{A_2,\mathbb{Z}_n\}$ is nonfinitely based and an explicit basis is: \begin{align} (xy)z=x(yz), \quad x^2 = x^{n + 2}, \quad xyx = x (yx)^{n + 1}, \quad xyxzx = xzxyx, \\ (x_1^n x_2^n \cdots x_k^n)^{3} = (x_1^n x_2^n \cdots x_k^n)^{2}, \quad k=2,3,\ldots \end{align}
As for nonfinitely generated varieties, apart from the example of Perkins (1969), there is an easy to describe example by J. R. Isbell (1970): the variety of monoids defined by $$ (x^py^p)^2 = (y^px^p)^2, \quad p = 2,3,5,7,11,\ldots \text{(primes)} $$
Pick a similarity type $\sigma$ that is "nice": finite with no zeroary operations will do. Consider the set B of all terms in the variables a and b, and for every such term $t(a,b)$ consider the identity that says this term is associative, or $t(a,t(b,c)) \approx t(t(a,b),c)$. If $t$ is $u(a)$, this is realized as $u(a) \approx u(u(a))$. Let $A_\sigma$ be the set of identities so produced.
In work which I distributed, I showed $A_\sigma$ is finitely based when $\sigma$ consists of a single binary operation. It remains finitely based if the type consists of only unary operations. What is also true (and should be published) is that for all other nice types, in particular the type with two binary operations, $A_\sigma$ is not finitely based.
More interesting facts about hyperidentities (such as hyperassociativity above) being represented by identities hold, but I have not kept up with the literature. I am confident one can manufacture many such examples of non finitely based varieties this way.
Gerhard "Should Get Back To Writing" Paseman, 2014.01.16

$\begingroup$ Thank you so much the answer is very helpful. I need this kind of examples to construct nonequational noetherian relatively free algebras. see the comment below for an example. $\endgroup$ – M. Shahryari Jan 16 '14 at 21:00

$\begingroup$ How can I find your preprint? I need to cite it in a paper which I am preparing. $\endgroup$ – M. Shahryari Jan 17 '14 at 7:03

$\begingroup$ For the moment, personal communication, I fear. You could ask Klaus Denecke, Shelly Wismath, or M. (Michal?) Kunc for the preprint they cite, A small basis for hyperassociativity, 1993, G. Paseman , U.C. Berkeley, but it is unlikely they have a copy, and if that had any nonfinite results, it would be based on ff=fff hyperidentity and unary types, derived from a MorseThue sequence. If I find a copy I will make it available. Kunc's thesis is on Citeseer, you might check it out. Gerhard "Maybe Should Finish My Writing" Paseman, 2014.01.17 $\endgroup$ – Gerhard Paseman Jan 17 '14 at 17:01

$\begingroup$ I shall edit this user profile to include my gmail address hints. If it is a short paper (less than 15 pages), I am willing to donate an hour skimming it and giving some critique. Gerhard "Proper Refereeing Would Cost More" Paseman, 2014.01.17 $\endgroup$ – Gerhard Paseman Jan 17 '14 at 17:08

$\begingroup$ Thank you so much, I uploaded a primary version of the paper today to Academia. It is 27 pages but you can read only the section 5. If you like, please go to my profile page and click on Academia link of mine. The name of paper is "Equational conditions in universal algebraic geometry". It is a join work of mine and my student. $\endgroup$ – M. Shahryari Jan 17 '14 at 19:37