Lokalno konačni varijeteti sa polu-distributivnom mrežom kongruencija
Locally finite varieties with semi{distributive congruence lattice.
Докторанд
Jovanović, JelenaМентор
Tanović, PredragЧланови комисије
Marković, PetarMijajlović, Žarko
Ikodinović, Nebojša
Tanović, Predrag
Метаподаци
Приказ свих података о дисертацијиСажетак
Predmet ove disertacije je sintaksna karakterizacija kongruencijske polu{distributiv-
nosti (u odnosu na inmum) lokalno konacnih varijeteta Maljcevljevim uslovima
(posmatramo varijetete idempotentnih algebri). Dokazujemo da takva karakteri-
zacija nije moguca sistemom identiteta koji koriste jedan ternarni i proizvoljan broj
binarnih operacijskih simbola. Prvu karakterizaciju dobijamo jakim Maljcevljevim
uslovom koji ukljucuje dva ternarna simbola: Lokalno konacan varijetet V zadovo-
ljava uslov kongruencijske polu{distributivnosti (u odnosu na inmum) ako i samo
ako postoje ternarni termi p i q (koji indukuju idempotentne term operacije) takvi
da V zadovoljava:
p(x; x; y) p(x; y; y)
p(x; y; x) q(x; y; x) q(x; x; y) q(y; x; x).
Ovaj uslov je optimalan u smislu da su broj terma, njihove visestrukosti i broj
identiteta najmanji moguci. Druga karakterizacija koju dobijamo koristi jedan 4-
arni simbol i data je jakim Maljcevljevim uslovom
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) :
Treca karakterizacija je data kompletnim Maljcevljevim uslovom: Postoje binarni
term t(x; y) i wnu-termi !n(x1; : : : ; xn) varijeteta V tako za sve n > 3 vazi sledece:
V j= !n(x; x; : : : ; x; y) t(x; y).
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 satises congruence
meet{semidistributivity if and only if there exist ternary terms p and q (inducing
idempotent term operations) such that V satises
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).