Example of a non-finitely based variety with explicit set of defining identities

Question

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

Answer 1

You can look at "Bases for Equational Theories of Semigroups" by P Perkins, J Algebra 11, 298-314 (1968). Theorem 2: the identities

$xyzw=xzyw$ and $yx^ky=xyx^{k-2}yx$ for $k=2,3,\dots$

define a non-finitely based variety of semigroups.

Answer 2

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]{c|ccccccc} 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 non-finitely 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 0-simple 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 non-finitely 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 non-finitely 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 $$ (x^py^p)^2 = (y^px^p)^2, \quad p = 2,3,5,7,11,\ldots \text{(primes)} $$

Answer 3

Pick a similarity type $\sigma$ that is "nice": finite with no zero-ary 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