Lokalno konačni varijeteti sa polu-distributivnom mrežom kongruencija

Abstract

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).