4 research outputs found
On power structures
In general, power structure of a structure A (with
universe A) is an appropriate structure defined on the power set
P(A). There are lot of papers concerning this topic in which power
structures appear explicitly or implicitly. The aim of this paper
is to give an overview of the results that are interesting from the
universal-algebraic point of view
Functional completeness and primitive positive decomposition of relations on finite domains
We give a new and elementary construction of primitive positive decomposition of higher arity relations into binary relations on finite domains. Such decompositions come up in applications to constraint satisfaction problems, clone theory and relational databases. The construction exploits functional completeness of 2-input functions in many-valued logic by interpreting relations as graphs of partially defined multivalued `functions'. The `functions' are then composed from ordinary functions in the usual sense. The construction is computationally effective and relies on well-developed methods of functional decomposition, but reduces relations only to ternary relations. An additional construction then decomposes ternary into binary relations, also effectively, by converting certain disjunctions into existential quantifications. The result gives a uniform proof of Peirce's reduction thesis on finite domains, and shows that the graph of any Sheffer function composes all relations there
Klonovi nedeterministiΔkih operacija
This thesis is a survey of some well-known and several new results concerning lattices of total, partial, incompletely specified clones and hyper-clones. We assign to every partial, incompletely specified and hyperoperation a suitable total operation and investigate thereby induced embeddings of the three lattices into corresponding lattices of total clones. Next we modify the famous Galois connection (Pol,Inv) between relations and operations for partial operations, IS operations and hyperoperations and describe classes of clones of IS operations and hyperoperations which strongly and weakly preserve given relations. We also state some known results concerning the four lattices on a two-element set. Finally, we present completeness criteria for the lattices of total and partial clones, and in the case of hyperclones and IS clones we describe four classes of coatoms, determined by four classes of Rosenbergβs relations.ΠΠ²Π° ΡΠ΅Π·Π° ΠΏΡΠ΅Π΄ΡΡΠ°Π²ΡΠ° ΠΏΡΠ΅Π³Π»Π΅Π΄ Π½Π΅ΠΊΠΈΡ
ΠΏΠΎΠ·Π½Π°ΡΠΈΡ
ΠΈ Π½Π΅ΠΊΠΎΠ»ΠΈΠΊΠΎ Π½ΠΎΠ²ΠΈΡ
ΡΠ΅Π·ΡΠ»ΡΠ°ΡΠ° Π²Π΅Π·Π°Π½ΠΈΡ
Π·Π° ΠΌΡΠ΅ΠΆΠ΅ ΡΠΎΡΠ°Π»Π½ΠΈΡ
, ΠΏΠ°ΡΡΠΈΡΠ°Π»Π½ΠΈΡ
, Π½Π΅ΠΏΠΎΡΠΏΡΠ½ΠΎ ΡΠΏΠ΅ΡΠΈΡΠΈΡΠΈΡΠ°Π½ΠΈΡ
ΠΊΠ»ΠΎΠ½ΠΎΠ²Π° ΠΈ Ρ
ΠΈΠΏΠ΅ΡΠΊΠ»ΠΎΠ½ΠΎΠ²Π°. Π‘Π²Π°ΠΊΠΎΡ ΠΏΠ°ΡΡΠΈΡΠ°Π»Π½ΠΎΡ, Π½Π΅ΠΏΠΎΡΠΏΡΠ½ΠΎ ΡΠΏΠ΅ΡΠΈΡΠΈΡΠΈΡΠ°Π½ΠΎΡ ΠΈ Ρ
ΠΈΠΏΠ΅ΡΠΎΠΏΠ΅ΡΠ°ΡΠΈΡΠΈ ΠΏΡΠΈΠ΄ΡΡΠΆΠΈΡΠ΅ΠΌΠΎ ΠΎΠ΄Π³ΠΎΠ²Π°ΡΠ°ΡΡΡΡ ΡΠΎΡΠ°Π»Π½Ρ ΠΎΠΏΠ΅ΡΠ°ΡΠΈΡΡ, ΠΈ ΠΈΡΠΏΠΈΡΡΡΠ΅ΠΌΠΎ ΡΠΈΠΌΠ΅ ΠΈΠ½Π΄ΡΠΊΠΎΠ²Π°Π½Π° ΠΏΠΎΡΠ°ΠΏΠ°ΡΠ° ΡΡΠΈ ΠΌΡΠ΅ΠΆΠ΅ Ρ ΠΎΠ΄Π³ΠΎΠ²Π°ΡΠ°ΡΡΡΠ΅ ΠΌΡΠ΅ΠΆΠ΅ ΡΠΎΡΠ°Π»Π½ΠΈΡ
ΠΊΠ»ΠΎΠ½ΠΎΠ²Π°. ΠΠΎΡΠΎΠΌ ΠΏΠΎΠ·Π½Π°ΡΡ ΠΠ°Π»ΠΎΠ°ΠΎΠ²Ρ Π²Π΅Π·Ρ (Pol,Inv) ΠΈΠ·ΠΌΠ΅ΡΡ ΡΠ΅Π»Π°ΡΠΈΡΠ° ΠΈ ΠΎΠΏΠ΅ΡΠ°ΡΠΈΡΠ° ΠΌΠΎΠ΄ΠΈΡΠΈΠΊΡΡΠ΅ΠΌΠΎ Π·Π° ΠΏΠ°ΡΡΠΈΡΠ°Π»Π½Π΅ ΠΎΠΏΠ΅ΡΠ°ΡΠΈΡΠ΅, ΠΠ‘ ΠΎΠΏΠ΅ΡΠ°-ΡΠΈΡΠ΅ ΠΈ Ρ
ΠΈΠΏΠ΅ΡΠΎΠΏΠ΅ΡΠ°ΡΠΈΡΠ΅ ΠΈ ΠΎΠΏΠΈΡΡΡΠ΅ΠΌΠΎ ΠΊΠ»Π°ΡΠ΅ ΠΊΠ»ΠΎΠ½ΠΎΠ²Π° Π½Π΅ΠΏΠΎΡΠΏΡΠ½ΠΎ ΡΠΏΠ΅ΡΠΈΡΠΈΡΠΈ-ΡΠ°Π½ΠΈΡ
ΠΈ Ρ
ΠΈΠΏΠ΅ΡΠΎΠΏΠ΅ΡΠ°ΡΠΈΡΠ° ΠΊΠΎΡΠ΅ ΡΠ°ΠΊΠΎ ΠΈ ΡΠ»Π°Π±ΠΎ ΡΡΠ²Π°ΡΡ Π΄Π°ΡΠ΅ ΡΠ΅Π»Π°ΡΠΈΡΠ΅. Π’Π°ΠΊΠΎΡΠ΅ Π½Π°Π²ΠΎΠ΄ΠΈΠΌΠΎ Π½Π΅ΠΊΠ΅ ΠΏΠΎΠ·Π½Π°ΡΠ΅ ΡΠ΅Π·ΡΠ»ΡΠ°ΡΠ΅ ΠΎ ΠΌΡΠ΅ΠΆΠ°ΠΌΠ° Π½Π° Π΄Π²ΠΎΠ΅Π»Π΅ΠΌΠ΅Π½ΡΠ½ΠΎΠΌ ΡΠΊΡΠΏΡ. ΠΠΎΠ½Π°ΡΠ½ΠΎ, Π½Π°Π²ΠΎΠ΄ΠΈΠΌΠΎ ΠΊΡΠΈΡΠ΅ΡΠΈΡΡΠΌΠ΅ ΠΊΠΎΠΌΠΏΠ»Π΅ΡΠ½ΠΎΡΡΠΈ Π·Π° ΠΌΡΠ΅ΠΆΠ΅ ΡΠΎΡΠ°Π»Π½ΠΈΡ
ΠΈ ΠΏΠ°ΡΡΠΈΡΠ°Π»Π½ΠΈΡ
ΠΊΠ»ΠΎΠ½ΠΎΠ²Π°, Π° Ρ ΡΠ»ΡΡΠ°ΡΡ Ρ
ΠΈΠΏΠ΅ΡΠΊΠ»ΠΎΠ½ΠΎΠ²Π° ΠΈ ΠΠ‘ ΠΊΠ»ΠΎΠ½ΠΎΠ²Π° ΠΎΠΏΠΈΡΡΡΠ΅ΠΌΠΎ ΡΠ΅ΡΠΈΡΠΈ ΠΊΠ»Π°ΡΠ΅ ΠΊΠΎΠ°ΡΠΎΠΌΠ°, ΠΎΠ΄ΡΠ΅ΡΠ΅Π½ΠΈΡ
ΡΠ° ΡΠ΅ΡΠΈΡΠΈ ΠΊΠ»Π°ΡΠ΅ Π ΠΎΠ·Π΅Π½Π±Π΅ΡΠ³ΠΎΠ²ΠΈΡ
ΡΠ΅Π»Π°ΡΠΈΡΠ°.Ova teza predstavlja pregled nekih poznatih i nekoliko novih rezultata vezanih za mreΕΎe totalnih, parcijalnih, nepotpuno specificiranih klonova i hiperklonova. Svakoj parcijalnoj, nepotpuno specificiranoj i hiperoperaciji pridruΕΎijemo odgovarajuΔu totalnu operaciju, i ispitujemo time indukovana potapanja tri mreΕΎe u odgovarajuΔe mreΕΎe totalnih klonova. Potom poznatu Galoaovu vezu (Pol,Inv) izmeΔu relacija i operacija modifikujemo za parcijalne operacije, NS opera-cije i hiperoperacije i opisujemo klase klonova nepotpuno specifici-ranih i hiperoperacija koje jako i slabo Δuvaju date relacije. TakoΔe navodimo neke poznate rezultate o mreΕΎama na dvoelementnom skupu. KonaΔno, navodimo kriterijume kompletnosti za mreΕΎe totalnih i parcijalnih klonova, a u sluΔaju hiperklonova i NS klonova opisujemo Δetiri klase koatoma, odreΔenih sa Δetiri klase Rozenbergovih relacija