Homogeneous and strictly homogeneous criteria for partial structures

Abstract

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 ...

Positive Primitive Structures

“…Then from Lemma 2.4 [10] we obtain that it is equivalent to: InvAutp(S)=InvAut(M). Next from Lemma 2.1 [10] we have InvAutp(S)= ) ( …”

“…This notion resembles the notion of strict (ultra) homogeneity with respect to finite partial automorphisms and total automorphisms of ℜ (see [3] and [10]). Namely, a structure is strictly homogeneous [10], if every f.p.a.…”

“…Notice that the property "InvAut(M) coincides with the set of all relations formed by first order formulas over a countable structure M" is equivalent to ω-categoricity of M (see [10]). Similarly we suggest that the full description of countable p.p.…”

“…Hence InvPol(M) coincides with the p.p. formula closure on M. On the other hand, since M is ω-categorical there is only a finite number of distinct predicates from InvAut(M) for any finite arity (e.g., see [10]). From InvPolp(S)⊆InvPol(S)⊆ InvPol(M)⊆InvAut(M) we obtain that the same is true for the set InvPolp(S).…”

“…Namely, a structure is strictly homogeneous [10], if every f.p.a. can be extended to an automorphism of this structure, also a relational structure with such property is called ultrahomogeneous [3].…”