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Abstract:
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The subject of this dissertation is a syntactic characterization of congruence ^{
semidistributivity in locally nite varieties by Mal'cev conditions (we consider va-
rieties of idempotent algebras). We prove that no such characterization is possible
by a system of identities including one ternary and any number of binary opera-
tion symbols. The rst characterization is obtained by a strong Mal'cev condition
involving two ternary term symbols: A locally nite variety V satis es congruence
meet{semidistributivity if and only if there exist ternary terms p and q (inducing
idempotent term operations) such that V satis es
p(x; x; y) p(x; y; y)
p(x; y; x) q(x; y; x) q(x; x; y) q(y; x; x).
This condition is optimal in the sense that the number of terms, their arities and
the number of identities are the least possible. The second characterization that we
nd uses a single 4-ary term symbol and is given by the following strong Mal'cev
condition
t(y; x; x; x) t(x; y; x; x) t(x; x; y; x)
t(x; x; x; y) t(y; y; x; x) t(y; x; y; x) t(x; y; y; x) :
The third characterization is given by a complete Mal'cev condition: There exist
a binary term t(x; y) and wnu-terms !n(x1; : : : ; xn) of variety V such that for all
n > 3 the following holds:
V j= !n(x; x; : : : ; x; y) t(x; y). |