A series of maximum subalgebras of direct products of algebras of finite-valued logics

Romov, B. A.

doi:10.1007/bf01069983

Cited by 11 publications

(30 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(12) Now using stipulation (9) we prove (11)⇔ (12) (substituting L ωω (P) by L ω0 (P) in (12) we get the identity and conversely replacing L ω 1 0 (P) by L ω0 (P) in (12) we obtain L ω 1 0 (L ωω (P)) = L ω0 (P) and hence L ωω (P) ⊆ L ω0 (P). Therefore, (11) holds).…”

Section: Finite Signature Relational Structuresmentioning

confidence: 92%

“…, n)) does not change the computational complexity of CSP(R). Thus, we come to utilize the Galois connections between primitive-positive definable sets of multisorted relations and n-clones of n-dimensional vectors of algebraic operations defined on the same base sets, which were established by the author in 1973 (e.g., see [12]). Note that unlike the conventional case, we still do not have any description of non-trivial minimal n-clones, even for the Boolean base sets (minimal clones are pivotal to discovering the borderline between tractability and intractability of CSP on a finite base set).…”

Section: Constraint Satisfaction Problemsmentioning

confidence: 99%

“…..,n )-formulas on a set of multi-sorted relations that corresponds to n-clones of vectors of algebraic operations (see [12]);…”

Section: Constraint Satisfaction Problemsmentioning

confidence: 99%

See 2 more Smart Citations

Homogeneous and strictly homogeneous criteria for partial structures

Romov¹

2009

Discrete Applied Mathematics

View full text Add to dashboard Cite

Language with finite string of quantifiers Homogeneous structure a b s t r a c tThe Galois closure on the set of relations invariant to all finite partial automorphisms (automorphisms) of a countable partial structure is established via quantifier-free infinite predicate languages (infinite languages with finite string of quantifiers respectively). Based on it the homogeneous and strictly homogeneous criteria for a countable partial structure as well as an ultrahomogeneous criterion for a countable relational structure are found. Next it is shown that infinite languages with a finite string of quantifiers cannot determine the corresponding Galois closure for relations invariant to all automorphisms of an uncountable partial structure.In this paper we explore two types of Galois connections between arbitrary sets of finite arity relations on an infinite base set E and sets of partial and total automorphisms of those relations. For finite partial automorphisms, the Galois closure on the relations side is determined by an infinite predicate language L α + 0 (card E = α), i.e., quantifier-free language with intersections of at most α predicates of the same arity. For countable E and total automorphisms we utilize the language L ω 1 ω , i.e., the language with countable intersections and a finite string of quantifiers.We apply these results to partial structures, having partial (both not everywhere defined and everywhere defined) operations in their signatures which also includes the case of conventional structures. This extension does not provide much new insight into classic systems which have everywhere defined main operations. Rather it may be helpful in areas of Systems Analysis where it is important to consider not everywhere defined operations either due to lack of information or due to the specifics of a problem, e.g., relational database inquiry functions.Next using the above Galois closures we represent semantic properties of countable partial structures with arbitrary signatures: to be ω-categorical, to be homogeneous and to be strictly homogeneous (which in the case of countable relational structures coincides with ultrahomogeneity [5]) via the conditions on their infinite predicate languages. Note that most of the criteria in this field are restricted to countable signatures (e.g., see [5,18,19]). The classification of strictly homogeneous countable structures with a single binary relation in its signature (in graphtheoretical interpretation: digraphs, graphs, tournaments etc.) is an established branch of infinite combinatorics (see [4,9,17]). However, there are no such results beyond the case of a binary relation which underscores the difficulty of the problem.The Galois connections, which are determined by fragments of finite predicate languages on one side and polymorphisms (algebraic operations which preserve atomic formulas) on the other, became an important tool in investigating Constraint Satisfaction Problems (CSP) that represent combinatorial algorithms. The basic connection is described via the ...

show abstract

Section: Finite Signature Relational Structuresmentioning

confidence: 92%

Section: Constraint Satisfaction Problemsmentioning

confidence: 99%

See 1 more Smart Citation

Homogeneous and strictly homogeneous criteria for partial structures

Romov¹

2009

Discrete Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…The case P 2 x ... x PI x Pk is considered in [8] and the case P 3 x P 3 is completely investigated in [3]. In addition, the papers [3,4,[6][7][8][9] contain a series of assertions on precomplete classes in systems P^ x . .…”

Section: Introductionmentioning

confidence: 99%

“…. x PI all precomplete classes were found in [9] (an alternative description is given in [6,7]). The case P 2 x ... x PI x Pk is considered in [8] and the case P 3 x P 3 is completely investigated in [3].…”

Section: Introductionmentioning

confidence: 99%