An algebra whose equational theory is finitely based but whose quasiequational theory is not, and vice versa.

Question

Is there an algebraic structure $K$ such that its equational theory has a finite basis, but that its quasiequational theory does not have a finite basis? Also, what about vice versa? That is, is there an algebraic structure $K$ such that its equational theory does not have a finite basis, but its quasiequational theory does have a finite basis?

Answer 1

Any nilpotent semigroup is finitely based. (If it is $k$-step nilpotent, then it satisfies the associative law and the $k$-step nilpotent law. Any further law is equivalent to one of the form $w_1(x_1,\ldots,x_k)\approx w_2(y_1,\ldots,y_k)$ in the fixed set of variables $x_1,y_1,\ldots,x_k, y_k$, and there are only finitely many inequivalent laws in these variables relative to the laws defining the variety of $k$-step nilpotent semigroups.)

On the other hand, a finite nilpotent semigroup with a finitely axiomatizable quasiequational theory must be a zero (or null) semigroup according to Corollary 1.4 of

RELATIVELY INHERENTLY NONFINITELY Q-BASED SEMIGROUPS
MARCEL JACKSON AND MIKHAIL VOLKOV
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 361, Number 4, April 2009, 2181-2206

(A null semigroup is one satisfying $x_1x_2\approx y_1y_2$.) So every finite, non-null, nilpotent semigroup has a finitely axiomatizable equational theory and a nonfinitely axiomatizable quasiequational theory.

For the other part of this question, in Theorem 7.10 of

FINITE BASIS THEOREMS FOR RELATIVELY CONGRUENCE-DISTRIBUTIVE QUASIVARIETIES
DON PIGOZZI
TRANSACTIONS OF THE AMERICAN MATHEMATICAL Volume 310, Number 2, December 1988, 499-533

one finds a procedure for starting with any finite algebra $A$ whose equational theory is not finitely axiomatizable, and constructing a new finite algebra, Et$(A)$, whose equational theory is also not finitely axiomatizable, but whose quasiequational theory is guaranteed to be finitely axiomatizable. If $A$ is Murskii's groupoid or Lyndon's groupoid, then Et$(A)$ provides the necessary example for the second part.