Forum Problem Lists Definitions FAQ

2.41 (1966)

Solved

Is the variety generated by
$\qquad$ a) a finite associative ring;
$\qquad$ b) a finite Lie ring;
$\qquad$ c) a finite quasigroup
finitely axiomatizable?
$\qquad$ d) What is the cardinality $n$ of the smallest semigroup generating a non-finitely axiomatizable variety?

Contributor: A. I. Mal’cev

Progress

a) Yes (I. V. L’vov, Algebra and Logic, 12 (1973), 156–167; R. L. Kruse, J. Algebra, 26 (1973), 298–318).

b) Yes (Yu. A. Bakhturin, A. Yu. Olshanskii, Math. USSR–Sb., 25 (1975), 507–523).

c) Not always (M. R. Vaughan-Lee, Algebra Universalis, 9 (1979), 269–280).

d) It is proved that $n = 6$ (P. Perkins, J. Algebra, 11 (1969), 298–314; A. N. Trakhtman, Semigroup Forum, 27 (1983), 387–389).

Comments

All comments are the responsibility of the user. Comments appearing on this page are not verified for correctness. Please keep posts mathematical and on topic. If you want to submit a proof (or a partial proof), please use the dedicated proof submission form rather than posting it in the comments.
Order by newest first or oldest first.

No comments yet. Be the first to comment.

Proof claims

Proof claims are the responsibility of the submitter. Appearance here does not mean the claim has been checked for mathematical correctness. Moderators only screen for spam, abuse, and obviously low-effort submissions.

No proof claims yet.