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.

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 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.

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.

scholarly journals A study on orderings based on fuzzy quasi- order relations

2021 ◽  
Author(s):  
Zhonglin Chai
Keyword(s):  
Partial Order ◽  
Order Relation ◽  
Fuzzy Relation ◽  
Partial Order Relation ◽  
Fuzzy Relations ◽  
Fuzzy Graph ◽  
Feasible Method ◽  
Order Relations ◽  
Finite Set ◽  
Quasi Order

Abstract This paper further studies orderings based on fuzzy quasi-order relations using fuzzy graph. Firstly, a fuzzy relation on a finite set is represented equivalently by a fuzzy graph. Using the graph, some new results on fuzzy relations are derived. In ranking those alternatives, we usually obtain a quasi-order relation, which often has inconsistencies, so it cannot be used for orderings directly. We need to remake it into a reasonable partial order relation for orderings. This paper studies these inconsistencies, and divides them into two types: framework inconsistencies and degree inconsistencies. For the former, a reasonable and feasible method is presented to eliminate them. To eliminate the latter, the concept of complete partial order relation is presented, which is more suitable than partial order relation to rank the alternatives. A method to obtain a reasonable complete partial order relation for a quasi-order relation is given also. An example is given as well to illustrate these discussions. Lastly, the paper discusses the connection between quasi-order relations and preference relations for orderings and some other related problems.

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.

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.

scholarly journals Critical Properties and Complexity Measures of Read-Once Boolean Functions

2021 ◽  
Author(s):  
Vadim Lozin ◽  
Mikhail Moshkov
Keyword(s):  
Boolean Functions ◽  
Critical Properties ◽  
Complexity Measures ◽  
Well Quasi Order ◽  
Finite Set ◽  
Quasi Order ◽  
Parameter Measuring ◽  
Two Parameters

AbstractIn this paper, we define a quasi-order on the set of read-once Boolean functions and show that this is a well-quasi-order. This implies that every parameter measuring complexity of the functions can be characterized by a finite set of minimal subclasses of read-once functions, where this parameter is unbounded. We focus on two parameters related to certificate complexity and characterize each of them in the terminology of minimal classes.

COUNTING THE RELATIONS COMPATIBLE WITH AN ALGEBRA

2010 ◽  
Vol 20 (07) ◽  
pp. 901-922 ◽  
Author(s):  
BRIAN A. DAVEY ◽  
JANE G. PITKETHLY
Keyword(s):  
Finite Number ◽  
Duality Theory ◽  
Necessary Conditions ◽  
Finite Algebra ◽  
Finiteness Condition ◽  
Natural Equivalence

We investigate when a finite algebra admits only a finite number of compatible relations (modulo a natural equivalence). This finiteness condition is closely related to others in the literature, and arises naturally in duality theory. We find necessary conditions for a finite algebra to admit only finitely many compatible relations, as well as a family of examples of such algebras.

scholarly journals Reverse mathematics and the equivalence of definitions for well and better quasi-orders

2004 ◽  
Vol 69 (3) ◽  
pp. 683-712 ◽  
Author(s):  
Peter Cholak ◽  
Alberto Marcone ◽  
Reed Solomon
Keyword(s):  
Reverse Mathematics ◽  
Second Order ◽  
Base System ◽  
Mathematical Concepts ◽  
Second Order Arithmetic ◽  
Finite Set ◽  
Combinatorial Principles ◽  
Quasi Order ◽  
Order Arithmetic ◽  
Better Quasi Order

In reverse mathematics, one formalizes theorems of ordinary mathematics in second order arithmetic and attempts to discover which set theoretic axioms are required to prove these theorems. Often, this project involves making choices between classically equivalent definitions for the relevant mathematical concepts. In this paper, we consider a number of equivalent definitions for the notions of well quasi-order and better quasi-order and examine how difficult it is to prove the equivalences of these definitions.As usual in reverse mathematics, we work in the context of subsystems of second order arithmetic and take RCA0 as our base system. RCA0 is the subsystem formed by restricting the comprehension scheme in second order arithmetic to formulas and adding a formula induction scheme for formulas. For the purposes of this paper, we will be concerned with fairly weak extensions of RCA0 (indeed strictly weaker than the subsystem ACA0 which is formed by extending the comprehension scheme in RCA0 to cover all arithmetic formulas) obtained by adjoining certain combinatorial principles to RCA0. Among these, the most widely used in reverse mathematics is Weak König's Lemma; the resulting theory WKL0 is extensively documented in [11] and elsewhere.We give three other combinatorial principles which we use in this paper. In these principles, we use k to denote not only a natural number but also the finite set {0, …, k − 1}.

Sign in / Sign up

Export Citation Format

Share Document