Smallest cardinality of finite algebraic structure whose quasi-equational theory is not finitely based?

Question

I understand that, for the set $\{0,1\}$ and any finite number of constants and operations on that set, the resulting algebraic structure has a finite basis of identities. But, does there exist a finite algebraic structure on $\{0,1\}$ whose quasi-equational theory is not finitely based? If not, is there such a structure on $\{0,1,2\}$? What is the smallest cardinality of a possible example?

Answer 1

does there exist a finite algebraic structure on {0,1} whose quasi-equational theory is not finitely based?

No, there does not exist a $2$-element algebra in a finite signature that has a nonfinitely axiomatizable quasiequational theory. This is the content of the main result of

V. A. Gorbunov
Quasi-identities of two-element algebras
Algebra and Logic 22, 83-88 (1983)

which states

THEOREM. Each two-element algebra of finite signature generates a minimal finitely based quasivariety.

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If not, is there such a structure on {0,1,2}?

Yes. In 1977, Gorbunov published an example of a $3$-element algebra with $2$ unary operations that does not have a finitely axiomatizable quasiequational theory. Later, in 1980, Mark Sapir published a proof that the $3$-element, 1-generated semigroup $C_{3,1} = \{x, x^2, x^3=x^4\}$ does not have a finitely axiomatizable quasiequational theory.

V. A. Gorbunov
Covers in lattices of quasivarieties and the independent axiomatizability.
Algebra i Logika 16(5):507-548, 1977.

M. V. Sapir
On the quasivarieties generated by finite semigroups
Semigroup Forum 20(1):73-88, 1980