Elementary equivalence of free algebras of varieties
by
Alexander Pinus
Novosibirsk State Technical University

As is known all infinitely generated free algebras of all varieties are elementary equivalent. Another situation we have for derived structures of this algebras: semigroups of endomorphisms End F_V(k), groups of automophisms Aut F_V(k), lattices of subalgebras Sub F_V(k) and the lattices of congrueces ConF_V(k). Here F_V(k) is k-generated V-free algebra for some variety V. So, this relation of elementary equivalence of derived structures of those free algebras gives some basic for some classification of infinitely generated V-free algebras for any variety V. This relation the following equivalence relations on the class of all infinite cardinals:

k \equiv _End^V\lambda <===> End F_V(k) \equiv EndF_V(\lambda),

k \equiv _Aut^V\lambda <===> Aut F_V(k) \equiv AutF_V(\lambda),

k \equiv _Sub^V\lambda <===> Sub F_V(k) \equiv AutF_V(\lambda),

k \equiv _Con^V\lambda <===> Con F_V(k) \equiv AutF_V(\lambda).

S.Shelah (1974) proved that the relation \equiv _End^V coincides with the relation \equiv ₂ for any nontrivial variety V. Here \equiv ₂ is relation of equivalence of cardinals in full second order logic. Let us denote the as \bigtriangledown and \equiv _p the following relations on the class of all infinite cardinals:

K\bigtriangledown\lambda <===> for any infinite cardinals K and \lambda,

K \equiv _p\lambda <===> the cardinals K and \lambda are equivalent in the second order logic with the quantifiers for any permutation on this cardinals.

In all known cases, the relations \equiv _Aut^V, \equiv _Sub^V, \equiv _Con^V coinside with either relation: \bigtriangledown, \equiv _p or \equiv ₂. Here, we give a survey of the known results for relations \equiv _Aut^V, \equiv _Sub^V, \equiv _Con^V for some varieties V, we formulate also some natural problems which are connected with this relations and formulate the following new results.

Theorem 1 For any nontrivial variety V of groups the lattice Con F_V(k) (Sub F_V(k)), is elementary definable in the class of all this lattices iff the cardinal k is definable in the full second order logic.

Theorem 2 For any variety V of unars the relations \equiv _Sub^V and \bigtriangledown are the same.

Theorem 3 For the variety V of all semigroups (of all commutative semigroups) the relations \equiv _Con^V, \equiv _Sub^V, \equiv ₂ are the same.

Theorem 4 For the variety V of all semilattices the relations \equiv _Con^V and \equiv ₂ are the same.

Date received: December 30, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caig-19.