HOW FINITE IS A THREE-ELEMENT UNARY ALGEBRA?

2005 ◽  
Vol 15 (02) ◽  
pp. 217-254 ◽  
Author(s):  
J. HYNDMAN ◽  
J. G. PITKETHLY
Keyword(s):  
Finite Rank ◽  
Duality Theory ◽  
Equational Theory ◽  
Unary Algebra ◽  
Natural Duality ◽  
Finitely Based ◽  
Injectivity Condition ◽  
Tight Connection ◽  
Algebraic Operations

We show that, within the class of three-element unary algebras, there is a tight connection between a finitely based quasi-equational theory, finite rank, enough algebraic operations (from natural duality theory) and a special injectivity condition.

scholarly journals POSITIVE FRAGMENTS OF RELEVANCE LOGIC AND ALGEBRAS OF BINARY RELATIONS

2010 ◽  
Vol 4 (1) ◽  
pp. 81-105 ◽  
Author(s):  
ROBIN HIRSCH ◽  
SZABOLCS MIKULÁS
Keyword(s):  
Equational Theory ◽  
Relevance Logic ◽  
Binary Relations ◽  
Finitely Based ◽  
Similarity Type ◽  
Relation Composition

We prove that algebras of binary relations whose similarity type includes intersection, union, and one of the residuals of relation composition form a nonfinitely axiomatizable quasivariety and that the equational theory is not finitely based. We apply this result to the problem of the completeness of the positive fragment of relevance logic with respect to binary relations.

scholarly journals A proof of Zhil'tsov's theorem on decidability of equational theory of epigroups

2016 ◽  
Vol Vol. 17 no. 3 (Combinatorics) ◽  
Author(s):  
Inna Mikhaylova
Keyword(s):  
Finite Semigroup ◽  
Equational Theory ◽  
Unary Operation ◽  
Finitely Based ◽  
Infinite Basis ◽  
International Audience

International audience Epigroups are semigroups equipped with an additional unary operation called pseudoinversion. Each finite semigroup can be considered as an epigroup. We prove the following theorem announced by Zhil'tsov in 2000: the equational theory of the class of all epigroups coincides with the equational theory of the class of all finite epigroups and is decidable. We show that the theory is not finitely based but provide a transparent infinite basis for it.

Equational properties of fixed-point operations in cartesian categories: An overview

2019 ◽  
Vol 29 (06) ◽  
pp. 909-925
Author(s):  
Z Ésik
Keyword(s):  
Fixed Point ◽  
Equational Theory ◽  
Continuous Functions ◽  
Equivalence Classes ◽  
Finitely Based ◽  
Additive Structure ◽  
Complete Lattices ◽  
Finite Base ◽  
Language Equivalence ◽  
Context Free

AbstractSeveral fixed-point models share the equational properties of iteration theories, or iteration categories, which are cartesian categories equipped with a fixed point or dagger operation subject to certain axioms. After discussing some of the basic models, we provide equational bases for iteration categories and offer an analysis of the axioms. Although iteration categories have no finite base for their identities, there exist finitely based implicational theories that capture their equational theory. We exhibit several such systems. Then we enrich iteration categories with an additive structure and exhibit interesting cases where the interaction between the iteration category structure and the additive structure can be captured by a finite number of identities. This includes the iteration category of monotonic or continuous functions over complete lattices equipped with the least fixed-point operation and the binary supremum operation as addition, the categories of simulation, bisimulation, or language equivalence classes of processes, context-free languages, and others. Finally, we exhibit a finite equational system involving residuals, which is sound and complete for monotonic or continuous functions over complete lattices in the sense that it proves all of their identities involving the operations and constants of cartesian categories, the least fixed-point operation and binary supremum, but not involving residuals.

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.

PSEUDORECURSIVE VARIETIES OF SEMIGROUPS — II

2012 ◽  
Vol 22 (05) ◽  
pp. 1250042 ◽  
Author(s):  
BENJAMIN WELLS
Keyword(s):  
Equational Theory ◽  
Unary Operation ◽  
Finitely Based ◽  
New Techniques ◽  
Distinguished Element ◽  
A Chain ◽  
Rewriting Rules ◽  
Semigroup Varieties

Constructions that yield pseudorecursiveness in [I] (Int. J. Algebra Comput.6 (1996) 457–510) are extended in this article. Finitely based varieties of semigroups with increasingly strict expansions by additional unary operation symbols or individual constants are shown to have the pseudorecursive property: the equational theory is undecidable, but the subsets obtained by bounding the number of distinct variables are all recursive. The most stringent case considered here is the single unary operation or distinguished element. New techniques of stratified reducibility and interpretation via rewriting rules are employed to show the property inherits along a chain of theories. Pure semigroup varieties that are both finitely based and pseudorecursive will be discussed in a later paper.

scholarly journals Distributive bilattices from the perspective of natural duality theory

2015 ◽  
Vol 73 (2) ◽  
pp. 103-141 ◽  
Author(s):  
L. M. Cabrer ◽  
H. A. Priestley
Keyword(s):  
Duality Theory ◽  
Natural Duality

scholarly journals The Max-Plus Algebra of the Natural Numbers has no Finite Equational Basis

1999 ◽  
Vol 6 (33) ◽  
Author(s):  
Luca Aceto ◽  
Zoltán Ésik ◽  
Anna Ingólfsdóttir
Keyword(s):  
Equational Theory ◽  
Geometric Characterization ◽  
Independent Interest ◽  
Free Algebras ◽  
Finitely Based ◽  
Natural Numbers ◽  
Equational Basis ◽  
Stepping Stone

This paper shows that the collection of identities which hold in<br />the algebra N of the natural numbers with constant zero, and binary<br />operations of sum and maximum is not finitely based. Moreover, it<br />is proven that, for every n, the equations in at most n variables that<br />hold in N do not form an equational basis. As a stepping stone in<br />the proof of these facts, several results of independent interest are<br />obtained. In particular, explicit descriptions of the free algebras in the<br />variety generated by N are offered. Such descriptions are based upon<br />a geometric characterization of the equations that hold in N, which<br />also yields that the equational theory of N is decidable in exponential<br />time.

A Self-Dual Equational Basis for Boolean Algebras

1983 ◽  
Vol 26 (1) ◽  
pp. 9-12 ◽  
Author(s):  
R. Padmanabhan
Keyword(s):  
Boolean Algebra ◽  
Equational Theory ◽  
Boolean Algebras ◽  
Duality Mapping ◽  
Finitely Based ◽  
Equational Basis ◽  
Cosmic Order ◽  
Principle Of Duality

AbstractThe principle of duality for Boolean algebra states that if an identity ƒ = g is valid in every Boolean algebra and if we transform ƒ = g into a new identity by interchanging (i) the two lattice operations and (ii) the two lattice bound elements 0 and 1, then the resulting identity ƒ = g is also valid in every Boolean algebra. Also, the equational theory of Boolean algebras is finitely based. Believing in the cosmic order of mathematics, it is only natural to ask whether the equational theory of Boolean algebras can be generated by a finite irredundant set of identities which is already closed for the duality mapping. Here we provide one such equational basis.

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