UNCOUNTABLY MANY DUALISABLE ALGEBRAS

2011 ◽  
Vol 21 (05) ◽  
pp. 825-839 ◽  
Author(s):  
JANE G. PITKETHLY
Keyword(s):  
Finite Type ◽  
Duality Theory ◽  
Natural Duality ◽  
Type Problem ◽  
Finite Algebras ◽  
Finite Set ◽  
Term Functions

Fix a finite set M with at least three elements. We find uncountably many different clones on M, each of which is the clone of term functions of a strongly dualisable algebra. This provides a solution to the Finite Type Problem of natural duality theory: there are finite algebras that are dualisable but not via a structure of finite type.

THE LATTICE OF ALTER EGOS

2012 ◽  
Vol 22 (01) ◽  
pp. 1250007 ◽  
Author(s):  
BRIAN A. DAVEY ◽  
JANE G. PITKETHLY ◽  
ROSS WILLARD
Keyword(s):  
Duality Theory ◽  
Finite Algebra ◽  
Galois Connection ◽  
Natural Duality ◽  
Algebraic Lattice ◽  
Finite Set ◽  
Partial Algebras ◽  
Quasi Order ◽  
Basic Concepts

We introduce a new Galois connection for partial operations on a finite set, which induces a natural quasi-order on the collection of all partial algebras on this set. The quasi-order is compatible with the basic concepts of natural duality theory, and we use it to turn the set of all alter egos of a given finite algebra into a doubly algebraic lattice. The Galois connection provides a framework for us to develop further the theory of natural dualities for partial algebras. The development unifies several fundamental concepts from duality theory and reveals a new understanding of full dualities, particularly at the finite level.

scholarly journals Relation between two twisted inverse image pseudofunctors in duality theory

2014 ◽  
Vol 151 (4) ◽  
pp. 735-764 ◽  
Author(s):  
Srikanth B. Iyengar ◽  
Joseph Lipman ◽  
Amnon Neeman
Keyword(s):  
Finite Type ◽  
Duality Theory ◽  
Inverse Image ◽  
Base Change ◽  
Hochschild Homology ◽  
Fundamental Class ◽  
Compactly Supported ◽  
Canonical Map ◽  
Flat Base ◽  
Derived Functors

Grothendieck duality theory assigns to essentially finite-type maps $f$ of noetherian schemes a pseudofunctor $f^{\times }$ right-adjoint to $\mathsf{R}f_{\ast }$, and a pseudofunctor $f^{!}$ agreeing with $f^{\times }$ when $f$ is proper, but equal to the usual inverse image $f^{\ast }$ when $f$ is étale. We define and study a canonical map from the first pseudofunctor to the second. This map behaves well with respect to flat base change, and is taken to an isomorphism by ‘compactly supported’ versions of standard derived functors. Concrete realizations are described, for instance for maps of affine schemes. Applications include proofs of reduction theorems for Hochschild homology and cohomology, and of a remarkable formula for the fundamental class of a flat map of affine schemes.

Finite algebras of relations are representable on finite sets

1999 ◽  
Vol 64 (1) ◽  
pp. 243-267 ◽  
Author(s):  
H. Andréka ◽  
I. Hodkinson ◽  
I. Németi
Keyword(s):  
Simple Proof ◽  
Finite Algebra ◽  
Finite Sets ◽  
Base Property ◽  
Finite Algebras ◽  
Relational Structures ◽  
Combinatorial Theorem ◽  
Finite Base ◽  
Algebras Of Relations ◽  
Finite Set

AbstractUsing a combinatorial theorem of Herwig on extending partial isomorphisms of relational structures, we give a simple proof that certain classes of algebras, including Crs, polyadic Crs, and WA, have the ‘finite base property’ and have decidable universal theories, and that any finite algebra in each class is representable on a finite set.

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.

Maximal n-generated subdirect products

2016 ◽  
Vol 26 (01) ◽  
pp. 123-155
Author(s):  
Joel Berman
Keyword(s):  
Upper Bound ◽  
Finite Algebra ◽  
Equivalence Relations ◽  
Subdirectly Irreducible ◽  
Finite Algebras ◽  
Locally Finite Variety ◽  
Congruence Relations ◽  
Finite Set ◽  
Basic Parameters ◽  
Subdirectly Irreducible Algebras

For [Formula: see text] a positive integer and [Formula: see text] a finite set of finite algebras, let [Formula: see text] denote the largest [Formula: see text]-generated subdirect product whose subdirect factors are algebras in [Formula: see text]. When [Formula: see text] is the set of all [Formula: see text]-generated subdirectly irreducible algebras in a locally finite variety [Formula: see text], then [Formula: see text] is the free algebra [Formula: see text] on [Formula: see text] free generators for [Formula: see text]. For a finite algebra [Formula: see text] the algebra [Formula: see text] is the largest [Formula: see text]-generated subdirect power of [Formula: see text]. For every [Formula: see text] and finite [Formula: see text] we provide an upper bound on the cardinality of [Formula: see text]. This upper bound depends only on [Formula: see text] and these basic parameters: the cardinality of the automorphism group of [Formula: see text], the cardinalities of the subalgebras of [Formula: see text], and the cardinalities of the equivalence classes of certain equivalence relations arising from congruence relations of [Formula: see text]. Using this upper bound on [Formula: see text]-generated subdirect powers of [Formula: see text], as [Formula: see text] ranges over the [Formula: see text]-generated subdirectly irreducible algebras in [Formula: see text], we obtain an upper bound on [Formula: see text]. And if all the [Formula: see text]-generated subdirectly irreducible algebras in [Formula: see text] have congruence lattices that are chains, then we characterize in several ways those [Formula: see text] for which this upper bound is obtained.

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 Pairs of Disjoint $q$-element Subsets Far from Each Other

10.37236/1606 ◽  
2001 ◽  
Vol 8 (2) ◽  
Author(s):  
Hikoe Enomoto ◽  
Gyula O. H. Katona
Keyword(s):  
Type Problem ◽  
Similar Nature ◽  
The Family ◽  
Finite Set

Let $n$ and $q$ be given integers and $X$ a finite set with $n$ elements. The following theorem is proved for $n>n_0(q)$. The family of all $q$-element subsets of $X$ can be partitioned into disjoint pairs (except possibly one if $n\choose q$ is odd), so that $|A_1\cap A_2|+|B_1\cap B_2|\leq q$, $|A_1\cap B_2|+|B_1\cap A_2| \leq q$ holds for any two such pairs $\{ A_1,B_1\} $ and $\{ A_2,B_2\} $. This is a sharpening of a theorem in [2]. It is also shown that this is a coding type problem, and several problems of similar nature are posed.

scholarly journals A Skolem–Mahler–Lech Theorem for Iterated Automorphisms of K–algebras

2015 ◽  
Vol 67 (2) ◽  
pp. 286-314
Author(s):  
Jason P. Bell ◽  
Jeffrey C. Lagarias
Keyword(s):  
Finite Type ◽  
Positive Characteristic ◽  
Finite Union ◽  
Arithmetic Progressions ◽  
Algebraic Extension ◽  
Finitely Generated ◽  
Finite Set ◽  
Affine Scheme ◽  
Algebra Over A Field

AbstractIn this paper we prove a commutative algebraic extension of a generalized Skolem–Mahler– Lech theorem. Let A be a finitely generated commutative K–algebra over a field of characteristic 0, and let σ be a K–algebra automorphism of A. Given ideals I and J of A, we show that the set S of integers m such that J is a finite union of complete doubly infinite arithmetic progressions in m, up to the addition of a finite set. Alternatively, this result states that for an affine scheme X of finite type over K, an automorphism σ∊ 2 AutK(X), and Y and Z any two closed subschemes of X, the set of integers m with Y is as above. We present examples showing that this result may fail to hold if the affine scheme X is not of finite type, or if X is of finite type but the field K has positive characteristic.

The Finite Type Problem for unary algebras

2010 ◽  
Vol 76 (3-4) ◽  
pp. 359-370
Author(s):  
Jane G. Pitkethly
Keyword(s):  
Finite Type ◽  
Type Problem

A STRONG PROPERTY OF THE WEAK SUBALGEBRA LATTICE FOR LOCALLY FINITE ALGEBRAS OF FINITE TYPE

2013 ◽  
Vol 23 (01) ◽  
pp. 1-35
Author(s):  
KONRAD PIÓRO
Keyword(s):  
Finite Type ◽  
Locally Finite ◽  
Finite Algebras ◽  
Edge Disjoint ◽  
Subalgebra Lattice ◽  
Strong Property ◽  
Directed Hypergraphs ◽  
Algebraic Result ◽  
Directed Hypergraph

The aim of this paper is to show that the weak subalgebra lattice uniquely determines the subalgebra lattice for locally finite algebras of a fixed finite type. However, this algebraic result turns out to be a very particular case of the following hypergraph result (which is interesting itself): A total directed hypergraph D of finite type is uniquely determined, in the class of all the directed hypergraphs of this type, by its skeleton up to the orientation of some pairwise edge-disjoint directed hypercycles and hyperpaths. The skeleton of D is a hypergraph obtained from D by omitting the orientation of all edges.

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