Finite axiomatizability for equational theories of computable groupoids

1989 ◽  
Vol 54 (3) ◽  
pp. 1018-1022 ◽  
Author(s):  
Peter Perkins
Keyword(s):  
Binary Operation ◽  
Finite Algebra ◽  
Equational Theory ◽  
Natural Numbers ◽  
Finite Axiomatizability ◽  
Finite Algebras ◽  
Equational Theories ◽  
Primitive Recursive ◽  
Positive Integers ◽  
Recursive Set

A computable groupoid is an algebra ‹N, g› where N is the set of natural numbers and g is a recursive (total) binary operation on N. A set L of natural numbers is a computable list of computable groupoids iff L is recursive, ‹N, ϕe› is a computable groupoid for each e ∈ L, and e ∈ L whenever e codes a primitive recursive description of a binary operation on N.Theorem 1. Let L be any computable list of computable groupoids. The set {e ∈ L: the equational theory of ‹N, ϕe› is finitely axiomatizable} is not recursive.Theorem 2. Let S be any recursive set of positive integers. A computable groupoid GS can be constructed so that S is inifinite iff GS has a finitely axiomatizable equational theory.The problem of deciding which finite algebras have finitely axiomatizable equational theories has remained open since it was first posed by Tarski in the early 1960's. Indeed, it is still not known whether the set of such finite algebras is recursively (or corecursively) enumerable. McKenzie [7] has shown that this question of finite axiomatizability for any (finite) algebra of finite type can be reduced to that for a (finite) groupoid.

scholarly journals EQUATIONAL COMPLEXITY OF THE FINITE ALGEBRA MEMBERSHIP PROBLEM

2008 ◽  
Vol 18 (08) ◽  
pp. 1283-1319 ◽  
Author(s):  
GEORGE F. McNULTY ◽  
ZOLTÁN SZÉKELY ◽  
ROSS WILLARD
Keyword(s):  
Finite Automata ◽  
Finite Algebra ◽  
Equational Theory ◽  
Upper And Lower Bounds ◽  
Variety Of Algebras ◽  
Membership Problem ◽  
Finite Algebras ◽  
Shift Automorphism ◽  
Positive Integers ◽  
Nonfinitely Based

We associate to each variety of algebras of finite signature a function on the positive integers called the equational complexity of the variety. This function is a measure of how much of the equational theory of a variety must be tested to determine whether a finite algebra belongs to the variety. We provide general methods for giving upper and lower bounds on the growth of equational complexity functions and provide examples using algebras created from graphs and from finite automata. We also show that finite algebras which are inherently nonfinitely based via the shift automorphism method cannot be used to settle an old problem of Eilenberg and Schützenberger.

A Hierarchy on the Class of Primitive Recursive Ordinal Functions

1976 ◽  
Vol 28 (6) ◽  
pp. 1205-1209
Author(s):  
Stanley H. Stahl
Keyword(s):  
Natural Numbers ◽  
Recursive Functions ◽  
Set Functions ◽  
Primitive Recursive ◽  
Class Of Functions ◽  
Recursive Set

The class of primitive recursive ordinal functions (PR) has been studied recently by numerous recursion theorists and set theorists (see, for example, Platek [3] and Jensen-Karp [2]). These investigations have been part of an inquiry concerning a larger class of functions; in Platek's case, the class of ordinal recursive functions and in the case of Jensen and Karp, the class of primitive recursive set functions. In [4] I began to study PR in depth and this paper is a report on an attractive analogy between PR and its progenitor, the class of primitive recursive functions on the natural numbers (Prim. Rec).

scholarly journals SAFE RECURSIVE SET FUNCTIONS

2015 ◽  
Vol 80 (3) ◽  
pp. 730-762 ◽  
Author(s):  
ARNOLD BECKMANN ◽  
SAMUEL R. BUSS ◽  
SY-DAVID FRIEDMAN
Keyword(s):  
Growth Rate ◽  
Polynomial Growth ◽  
Turing Machines ◽  
Finite Sets ◽  
Infinite Time ◽  
Machine Model ◽  
Set Functions ◽  
Rate Functions ◽  
Primitive Recursive ◽  
Recursive Set

AbstractWe introduce the safe recursive set functions based on a Bellantoni–Cook style subclass of the primitive recursive set functions. We show that the functions computed by safe recursive set functions under a list encoding of finite strings by hereditarily finite sets are exactly the polynomial growth rate functions computed by alternating exponential time Turing machines with polynomially many alternations. We also show that the functions computed by safe recursive set functions under a more efficient binary tree encoding of finite strings by hereditarily finite sets are exactly the quasipolynomial growth rate functions computed by alternating quasipolynomial time Turing machines with polylogarithmic many alternations.We characterize the safe recursive set functions on arbitrary sets in definability-theoretic terms. In its strongest form, we show that a function on arbitrary sets is safe recursive if and only if it is uniformly definable in some polynomial level of a refinement of Jensen's J-hierarchy, relativized to the transitive closure of the function's arguments.We observe that safe recursive set functions on infinite binary strings are equivalent to functions computed by infinite-time Turing machines in time less than ωω. We also give a machine model for safe recursive set functions which is based on set-indexed parallel processors and the natural bound on running times.

Various varieties

1973 ◽  
Vol 16 (3) ◽  
pp. 363-367 ◽  
Author(s):  
Sheila Oates MacDonald
Keyword(s):  
Finite Algebra ◽  
Finite Basis ◽  
Finitely Based ◽  
Universal Algebras ◽  
Finite Algebras ◽  
The Past ◽  
Bounded Number

The study of varieties of universal algebras2 which was initiated by Birkhoff in 1935, [2], has received considerable attention during the past decade; the question of particular interest being: “Which varieties have a finite basis for their laws?” In that paper Birkhoff showed that the laws of a finite algebra which involve a bounded number of variables are finitely based, so it is not altogether surprising that finite algebras have received their share of this attention.

scholarly journals The Equation of the Set of Natural Numbers Just to Sum

2021 ◽  
Vol 8 (5) ◽  
pp. 379-388
Author(s):  
Tulus Nadapdap ◽  
Tulus . ◽  
Opim Salim
Keyword(s):  
Natural Number ◽  
Unique Solution ◽  
Systems Of Equations ◽  
Natural Numbers ◽  
Similar Construction ◽  
Language Equations ◽  
Letter Alphabet ◽  
Periodic Constant ◽  
Recursive Set

Systems of equations of the form X = Y + Z and X = C, in which the unknowns are sets of integers,”+” denotes pairwise sum of sets S + T = m + n m S, n T , and C is an ultimately periodic constant. When restricted to sets of natural numbers, such equations can be equally seen as language equations over a one-letter alphabet with concatenation and regular constants, and it is shown that such systems are computationally universal, in the sense that for every recursive set S N there exists a system with a unique solution containing T with S = n 16n + 13 T. For systems over sets of all integers, both positive and negative, there is a similar construction of a system with a unique solution S = {n|16n ∈ T} representing any hyper-arithmetical set S ⊆ N. Keywords: Language equations, Natural numbers, Equations of natural number.

scholarly journals Partitions of the Natural Numbers

1964 ◽  
Vol 7 (2) ◽  
pp. 219-236 ◽  
Author(s):  
Myer Angel
Keyword(s):  
Natural Numbers ◽  
Positive Integers

We obtain in this article some results concerning partitions of the natural numbers, the most important of which is a generalization of that quoted immediately below. Some intuitive material is included.In 1954, J. Lambek and L. Moser [l] showed that "Two non-decreasing sequences f and g (of non-negative integers) are inverses if and only if the corresponding sets F and G of positive integers, defined by F(m) = the mth element of F = f(m) + m and G(n) = g(n) + n are complementary."

Primitive recursive set functions

1971 ◽  
pp. 143-176 ◽  
Author(s):  
Ronald B. Jensen ◽  
Carol Karp
Keyword(s):  
Set Functions ◽  
Primitive Recursive ◽  
Recursive Set

A note on equational theories

2000 ◽  
Vol 65 (4) ◽  
pp. 1705-1712 ◽  
Author(s):  
Markus Junker
Keyword(s):  
Equational Theory ◽  
Order Theory ◽  
Point Of View ◽  
Unpublished Manuscript ◽  
Positive Information ◽  
Closed Sets ◽  
First Order Theory ◽  
Equational Theories ◽  
Definable Sets ◽  
Image Position

Several attempts have been done to distinguish “positive” information in an arbitrary first order theory, i.e., to find a well behaved class of closed sets among the definable sets. In many cases, a definable set is said to be closed if its conjugates are sufficiently distinct from each other. Each such definition yields a class of theories, namely those where all definable sets are constructible, i.e., boolean combinations of closed sets. Here are some examples, ordered by strength:Weak normality describes a rather small class of theories which are well understood by now (see, for example, [P]). On the other hand, normalization is so weak that all theories, in a suitable context, are normalizable (see [HH]). Thus equational theories form an interesting intermediate class of theories. Little work has been done so far. The original work of Srour [S1, S2, S3] adopts a point of view that is closer to universal algebra than to stability theory. The fundamental definitions and model theoretic properties can be found in [PS], though some easy observations are missing there. Hrushovski's example of a stable non-equational theory, the first and only one so far, is described in the unfortunately unpublished manuscript [HS]. In fact, it is an expansion of the free pseudospace constructed independently by Baudisch and Pillay in [BP] as an example of a strictly 2-ample theory. Strong equationality, defined in [Hr], is also investigated in [HS].

scholarly journals Polynomial density of commutative semigroups

1993 ◽  
Vol 48 (1) ◽  
pp. 151-162 ◽  
Author(s):  
Andrzej Kisielewicz ◽  
Norbert Newrly
Keyword(s):  
Commutative Semigroup ◽  
Equational Theory ◽  
Full Description ◽  
Commutative Semigroups ◽  
Equational Theories

An algebra is said to be polynomially n−dense if all equational theories extending the equational theory of the algebra with constants have a relative base consisting of equations in no more than n variables. In this paper, we investigate polynomial density of commutative semigroups. In particular, we prove that, for n > 1, a commutative semigroup is (n − 1)-dense if and only if its subsemigroup consisting of all n−factor-products is either a monoid or a union of groups of a bounded order. Moreover, a commutative semigroup is 0-dense if and only if it is a bounded semilattice. For semilattices, we give a full description of the corresponding lattices of equational theories.

Orders on magmas and computability theory

2018 ◽  
Vol 27 (07) ◽  
pp. 1841001
Author(s):  
Trang Ha ◽  
Valentina Harizanov
Keyword(s):  
Algebraic Structure ◽  
Knot Theory ◽  
Binary Operation ◽  
Computability Theory ◽  
Linear Ordering ◽  
Natural Numbers ◽  
Right Order

We investigate algebraic and computability-theoretic properties of orderable magmas. A magma is an algebraic structure with a single binary operation. A right order on a magma is a linear ordering of its domain, which is right-invariant with respect to the magma operation. We use tools of computability theory to investigate Turing complexity of orders on computable orderable magmas. A magma is computable if it is finite, or if its domain can be identified with the set of natural numbers and the magma operation is computable. Interesting orderable magmas that are not even associative come from knot theory.

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