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2.40 (1966)

Partially Solved

The $I$-theory ($Q$-theory) of a class $\mathfrak{K}$ of universal algebras is the totality of all identities (quasi-identities) that are valid on all the algebras in $\mathfrak{K}$. Does there exist a finitely axiomatizable variety of
$\quad$ a) groups,
$\quad$ b) semigroups,
$\quad$ c) $\ $(1) rings
$\quad\quad\ $ (2) of associative rings
$\quad\quad\qquad$ (i) whose $I$-theory is non-decidable?
$\quad\quad\qquad$ (ii) whose $Q$-theory is non-decidable?
$\quad\quad\ $ (3) of Lie rings
$\quad\quad\qquad$ (i) whose $I$-theory is non-decidable?
$\quad\quad\qquad$ (ii) whose $Q$-theory is non-decidable?

Contributor: A. I. Mal’cev

Progress

a) Yes (Yu. G. Kleiman, Trans. Moscow Math. Soc., 1983, no. 2, 63–110).

b) Yes (V. L. Murskiĭ, Math. Notes, 3 (1968), 423–427).

c) (1) Yes (V. Yu. Popov, Math. Notes, 67 (2000), 495–504). (2i) No, it does not (A. Ya. Belov, Izv. Math., 74, no. 1 (2010), 1–126). (2ii) and (3ii): Yes, it exists (A. I. Budkin, Izv. Altai Univ., 65, no. 1 (2010), 15–17 (Russian)).

c) (3) (i) It is not difficult to find a recursively axiomatizable variety of semigroups with identity whose $I$-theory is non-recursive (see also A. I. Mal’cev, Mat. Sbornik, 69, no. 1 (1966), 3–12 (Russian)).

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