scholarly journals DOMINIONS AND PRIMITIVE POSITIVE FUNCTIONS

2018 ◽  
Vol 83 (1) ◽  
pp. 40-54 ◽  
Author(s):  
MIGUEL CAMPERCHOLI
Keyword(s):  
Partial Function ◽  
Finite Algebras ◽  
Positive Formula ◽  
Primitive Positive Formula ◽  
Near Unanimity Term ◽  
Finite Set ◽  
Image Position ◽  
Positive Functions ◽  
Near Unanimity

AbstractLetA≤Bbe structures, and${\cal K}$a class of structures. An elementb∈BisdominatedbyArelative to${\cal K}$if for all${\bf{C}} \in {\cal K}$and all homomorphismsg,g':B → Csuch thatgandg'agree onA, we havegb=g'b. Our main theorem states that if${\cal K}$is closed under ultraproducts, thenAdominatesbrelative to${\cal K}$if and only if there is a partial functionFdefinable by a primitive positive formula in${\cal K}$such thatFB(a1,…,an) =bfor somea1,…,an∈A. Applying this result we show that a quasivariety of algebras${\cal Q}$with ann-ary near-unanimity term has surjective epimorphisms if and only if$\mathbb{S}\mathbb{P}_n \mathbb{P}_u \left( {\mathcal{Q}_{{\text{RSI}}} } \right)$has surjective epimorphisms. It follows that if${\cal F}$is a finite set of finite algebras with a common near-unanimity term, then it is decidable whether the (quasi)variety generated by${\cal F}$has surjective epimorphisms.

scholarly journals Finitely Related Algebras In Congruence Distributive Varieties Have Near Unanimity Terms

2013 ◽  
Vol 65 (1) ◽  
pp. 3-21 ◽  
Author(s):  
Libor Barto
Keyword(s):  
Constraint Satisfaction Problem ◽  
Distributive Variety ◽  
Term Operation ◽  
Relational Structures ◽  
Near Unanimity Term ◽  
Congruence Distributive Variety ◽  
Finite Set ◽  
Congruence Distributive ◽  
Near Unanimity ◽  
Finitely Related

Abstractwe show that every finite, finitely related algebra in a congruence distributive variety has a near unanimity term operation. as a consequence we solve the near unanimity problem for relational structures: it is decidable whether a given finite set of relations on a finite set admits a compatible near unanimity operation. this consequence also implies that it is decidable whether a given finite constraint language defines a constraint satisfaction problem of bounded strict width.

The syntax and semantics of entailment in duality theory

1995 ◽  
Vol 60 (4) ◽  
pp. 1087-1114 ◽  
Author(s):  
B. A. Davey ◽  
M. Haviar ◽  
H. A. Priestley
Keyword(s):  
Duality Theory ◽  
Finite Algebra ◽  
Partial Functions ◽  
Distributive Variety ◽  
Positive Formula ◽  
Primitive Positive Formula ◽  
Congruence Distributive Variety ◽  
Finite Set ◽  
Clone Theory ◽  
Congruence Distributive

AbstractBoth syntactic and semantic solutions are given for the entailment problem of duality theory. The test algebra theorem provides both a syntactic solution to the entailment problem in terms of primitive positive formula and a new derivation of the corresponding result in clone theory, viz. the syntactic description of Inv(Pol(R)) for a given set R of unitary relations on a finite set. The semantic solution to the entailment problem follows from the syntactic one, or can be given in the form of an algorithm. It shows, in the special case of a purely relational type, that duality-theoretic entailment is describable in terms of five constructs, namely trivial relations, intersection, repetition removal, product, and retractive projection. All except the last are concrete, in the sense that they are described by a quantifier-free formula. It is proved that if the finite algebra M generates a congruence-distributive variety and all subalgebras of M are subdirectly irreducible, then concrete constructs suffice to describe entailment. The concept of entailment appropriate to strong dualities is also introduced, and described in terms of coordinate projections, restriction of domains, and composition of partial functions.

scholarly journals CLONES OF ALGEBRAS WITH PARALLELOGRAM TERMS

2012 ◽  
Vol 22 (01) ◽  
pp. 1250005 ◽  
Author(s):  
KEITH A. KEARNES ◽  
ÁGNES SZENDREI
Keyword(s):  
Relational Clone ◽  
Near Unanimity Term ◽  
Finite Set ◽  
Near Unanimity

We describe a manageable set of relations that generates the finitary relational clone of an algebra with a parallelogram term. This result applies to any algebra with a Maltsev term and to any algebra with a near unanimity term. One consequence of the main result is that on any finite set and for any finite k there are only finitely many clones of algebras with a k-ary parallelogram term which generate residually small varieties.

scholarly journals Natural duality via a finite set of relations

1995 ◽  
Vol 51 (3) ◽  
pp. 469-478 ◽  
Author(s):  
László Zádori
Keyword(s):  
Duality Theorem ◽  
Finite Algebra ◽  
Natural Duality ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Term Operation ◽  
Near Unanimity Term ◽  
Finite Set ◽  
Necessary And Sufficient ◽  
Near Unanimity

We present a duality theorem. We give a necessary and sufficient condition for any set of algebraic relations to entail the set of all algebraic relations in Davey and Werner's sense. The main result of the paper states that for a finite algebra a finite set of algebraic relations yields a duality if and only if the set of all algebraic relations can be obtained from it by using four types of relational constructs. Finally, we prove that a finite algebra admits a natural duality if and only if the algebra has a near unanimity term operation, provided that the algebra possesses certain 2k-ary term operations for some k. This is a generalisation of a theorem of Davey, Heindorf and McKenzie.

Falconer's formula for the Hausdorff dimension of a self-affine set in R2

1995 ◽  
Vol 15 (1) ◽  
pp. 77-97 ◽  
Author(s):  
Irene Hueter ◽  
Steven P. Lalley
Keyword(s):  
Hausdorff Dimension ◽  
Dimensional Hausdorff Measure ◽  
Similarity Transformations ◽  
Affine Mappings ◽  
Finite Set ◽  
Self Similar ◽  
Image Position ◽  
Affine Set ◽  
Small Overlap ◽  
Box Dimensions

Let A1, A2,…,Ak be a finite set of contractive, affine, invertible self-mappings of R2. A compact subset Λ of R2 is said to be self-affine with affinitiesA1, A2,…,Ak ifIt is known [8] that for any such set of contractive affine mappings there is a unique (compact) SA set with these affinities. When the affine mappings A1, A2,…,Ak are similarity transformations, the set Λ is said to be self-similar. Self-similar sets are well understood, at least when the images Ai(Λ) have ‘small’ overlap: there is a simple and explicit formula for the Hausdorff and box dimensions [12, 10]; these are always equal; and the δ-dimensional Hausdorff measure of such a set (where δ is the Hausdorff dimension) is always positive and finite.

scholarly journals THE VARIETY OF COSET RELATION ALGEBRAS

2018 ◽  
Vol 83 (04) ◽  
pp. 1595-1609 ◽  
Author(s):  
STEVEN GIVANT ◽  
HAJNAL ANDRÉKA
Keyword(s):  
Large Class ◽  
Identity Element ◽  
Relation Algebra ◽  
Order Theory ◽  
Relation Algebras ◽  
First Order ◽  
First Order Theory ◽  
Atomic Pair ◽  
Finite Set ◽  
Image Position

AbstractGivant [6] generalized the notion of an atomic pair-dense relation algebra from Maddux [13] by defining the notion of a measurable relation algebra, that is to say, a relation algebra in which the identity element is a sum of atoms that can be measured in the sense that the “size” of each such atom can be defined in an intuitive and reasonable way (within the framework of the first-order theory of relation algebras). In Andréka--Givant [2], a large class of examples of such algebras is constructed from systems of groups, coordinated systems of isomorphisms between quotients of the groups, and systems of cosets that are used to “shift” the operation of relative multiplication. In Givant--Andréka [8], it is shown that the class of these full coset relation algebras is adequate to the task of describing all measurable relation algebras in the sense that every atomic and complete measurable relation algebra is isomorphic to a full coset relation algebra.Call an algebra $\mathfrak{A}$ a coset relation algebra if $\mathfrak{A}$ is embeddable into some full coset relation algebra. In the present article, it is shown that the class of coset relation algebras is equationally axiomatizable (that is to say, it is a variety), but that no finite set of sentences suffices to axiomatize the class (that is to say, the class is not finitely axiomatizable).

Degree Functions in Local Rings

1961 ◽  
Vol 57 (1) ◽  
pp. 1-7 ◽  
Author(s):  
D. Rees
Keyword(s):  
Maximal Ideal ◽  
Zero Element ◽  
Transcendence Degree ◽  
Local Rings ◽  
Local Domain ◽  
Degree Function ◽  
Finite Set ◽  
Residue Fields ◽  
Image Position ◽  
The Ideal

Let Q be a local domain of dimension d with maximal ideal m and let q be an m-primary ideal. Then we define the degree function dq(x) to be the multiplicity of the ideal , where x; is a non-zero element of m. The degree function was introduced by Samuel (5) in the case where q = m. The function dq(x) satisfies the simple identityThe main purpose of this paper is to obtain a formulawhere vi(x) denotes a discrete valuation centred on m (i.e. vi(x) ≥ 0 if x ∈ Q, vi(x) > 0 if x ∈ m) of the field of fractions K of Q. The valuations vi(x) are assumed to have the further property that their residue fields Ki have transcendence degree d − 1 over k = Q/m. The symbol di(q) denotes a non-negative integer associated with vi(x) and q which for fixed q is zero for all save a finite set of valuations vi(x).

The Orbit-Stabilizer Problem for Linear Groups

1985 ◽  
Vol 37 (2) ◽  
pp. 238-259 ◽  
Author(s):  
John D. Dixon
Keyword(s):  
Vector Space ◽  
Linear Group ◽  
General Linear Group ◽  
General Linear ◽  
Linear Groups ◽  
Rational Field ◽  
Finite Set ◽  
Image Position ◽  
Orbit Problem

Let G be a subgroup of the general linear group GL(n, Q) over the rational field Q, and consider its action by right multiplication on the vector space Qn of n-tuples over Q. The present paper investigates the question of how we may constructively determine the orbits and stabilizers of this action for suitable classes of groups. We suppose that G is specified by a finite set {x1, …, xr) of generators, and investigate whether there exist algorithms to solve the two problems:(Orbit Problem) Given u, v ∊ Qn, does there exist x ∊ G such that ux = v; if so, find such an element x as a word in x1, …, xr and their inverses.(Stabilizer Problem) Given u, v ∊ Qn, describe all words in x1, …, xr and their inverses which lie in the stabilizer

A generalization of McDiarmid's theorem for mixed Bernoulli percolation

1980 ◽  
Vol 88 (1) ◽  
pp. 167-170 ◽  
Author(s):  
J. M. Hammersley
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Finite Sequence ◽  
Random Set ◽  
Percolation Probability ◽  
Open Path ◽  
Finite Set ◽  
Image Position ◽  
Infinite Set ◽  
Mutually Independent

Let G be an infinite partially directed graph of finite outgoing degree. Thus G consists of an infinite set of vertices, together with a set of edges between certain prescribed pairs of vertices. Each edge may be directed or undirected, and the number of edges from (but not necessarily to) any given vertex is always finite (though possibly unbounded). A path on G from a vertex V1 to a vertex Vn (if such a path exists) is a finite sequence of alternate edges and vertices of the form E12, V2, E23, V3, …, En − 1, n, Vn such that Ei, i + 1 is an edge connecting Vi and Vi + 1 (and in the direction from Vi to Vi + 1 if that edge happens to be directed). In mixed Bernoulli percolation, each vertex Vi carries a random variable di, and each edge Eij carries a random variable dij. All these random variables di and dij are mutually independent, and take only the values 0 or 1; the di take the value 1 with probability p, while the dij take the value 1 with probability p. A path is said to be open if and only if all the random variables carried by all its edges and all its vertices assume the value 1. Let S be a given finite set of vertices, called the source set; and let T be the set of all vertices such that there exists at least one open path from some vertex of S to each vertex of T. (We imagine that fluid, supplied to all the source vertices, can flow along any open path; and thus T is the random set of vertices eventually wetted by the fluid). The percolation probabilityis defined to be the probability that T is an infinite set.

A selection theorem

1983 ◽  
Vol 48 (3) ◽  
pp. 585-594
Author(s):  
Lefteris Miltiades Kirousis
Keyword(s):  
Partial Function ◽  
Existential Quantifier ◽  
Selection Theorem ◽  
Unbounded Subset ◽  
Image Position ◽  
Existential Quantification

In [1978] Harrington and MacQueen proved that if B is an (A, E)-semirecursive subset of A, such that the functions in BA can be coded as elements of A in an (A, E)-recursive way, then ENV(A, E) is closed under the existential quantifier ∃T ∈ B.Later Moschovakis showed that if ENV(Vκ, ∈, E) is closed under the quantifier ∃t ∈ λ, where λ is the p-cofinality of κ, thenthe p-cofinality of κ is the least ordinal λ for which there exists a (κ, <, E)-recursive partial function ƒ into κ, such that ƒ∣λ is total from λ onto an unbounded subset of κ.In this paper we prove that for any infinite ordinal κ if p-card(κ) = κ, then ENV(κ, <, E) is closed under ∃t ∈ μ, for μ < p-cf(κ); p-cf(κ) is the “boldface” analog of p-cf((κ) and p-card(κ) is defined similarly.From this follows that for any infinite ordinal κ the following two statements are equivalent.(i) ENV(κ, <, E) is closed under bounded existential quantification.(ii) ENV(κ, <, E) = ENV(κ, <, E#) or p-cf(κ) = κ.We also show that we cannot omit any of the hypotheses in the above theorem.We follow mainly the notation of Kechris and Moschovakis [1977].

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